Sanyi Tang
email: sytang@snnu.edu.cn

Guangyao Tang

Robert A Cheke
email: r.a.cheke@greenwich.ac.uk

Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases

Keywords: Optimum timing, pest control, IPM, Economic threshold, Augmentation, Predator-prey model

Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest-natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided.

Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications.

Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial.

To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics.

A c c e p t e d m a n u s c r i p t 1 Introduction

Integrated pest management (IPM) involves choosing appropriate tactics from a range of pest control techniques including biological, cultural and chemical methods to suit individual cropping systems, pest complexes and local environments [START_REF] Flint | Integrated Pest Management for Walnuts, University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, Second Edition[END_REF]van Lenteren, 1995van Lenteren, , 2000;;van Lenteren and Woets, 1988). Biological control is often a component of an IPM strategy [START_REF] Greathead | Natural enemies of tropical locusts and grasshoppers: their impact and potential as biological control agents[END_REF]Parker, 1971). It is defined as the reduction of pest populations by natural enemies and typically involves an active human role, such as augmentation which involves the supplemental release of natural enemies. Relatively few natural enemies may be released at a critical time of the season (inoculative release) or millions may be released (inundative release) when insufficient reproduction of released natural enemies is likely to occur and pest control will be achieved exclusively by the released individuals themselves [START_REF] Hoffmann | Natural enemies of vegetable insect pests[END_REF]Neuenschwander and Herren, 1988).

Another important method for pest control is chemical control. In most cropping systems, insecticides are still the principal means of controlling pests once the economic threshold (ET) has been reached. They can be relatively cheap and are easy to apply, fast-acting, and in most instances can be relied on to control the pests [START_REF] Hoffmann | Natural enemies of vegetable insect pests[END_REF]). An ET is usually defined as the number of insect pests in the field when control actions must be taken to prevent the economic injury level from being reached and exceeded (Pedigo and Higley, 1992;van Lenteren, 1995), as shown in Fig. 1. However, only if all other IPM tactics including biological and cultural control are unable to keep an insect pest population below an ET, then use of an insecticide to control the pest and prevent economic loss is justified. Even where pesticides are included as a last resort as

A c c e p t e d m a n u s c r i p t

part of an IPM strategy, the emphasis is on rational use involving suitable active ingredient selection and careful timing of application with regard to pest population levels.

Most importantly, chemical and biological methods have often proven antagonistic pesticides limit populations of natural enemies in crops and reduce their impact on pest populations. In extreme cases of such antagonism, often called resurgence, pest populations may be increased by the application of insecticides which kill both the pest and their natural enemies [START_REF] Barclay | Models for pest control using predator release, habitat management and pesticide release in combination[END_REF][START_REF] Debach | Biological control by natural enemies[END_REF]Ruberson, 1998). So it is important to know how such antagonistic pesticides affect the outcomes of pest control measures.

One approach to understanding the range of possible ecological interactions between pest, natural enemy and pesticides is to construct and explore population models. [START_REF] Barclay | Models for pest control using predator release, habitat management and pesticide release in combination[END_REF], [START_REF] Barlow | A model for the effect of Sphecophaga vesparum as a biological control agent of the common wasp in New Zealand[END_REF] and [START_REF] Barclay | Predator-prey models with added mortality[END_REF] have developed continuous ODE models, with coupled predator and pest populations, both of which are affected by insecticides. [START_REF] Carpenter | Effect of control measures on pest populations subject to regulation by parasites and pathogens[END_REF] has developed a similar model for the interaction between pests, pathogens and insecticides. Recently, continuous or discrete predator-prey models concerning IPM strategies have been developed and investigated (Tang and Chen, 2004;Tang and Cheke, 2005;Tang et al., 2005;Tang and Cheke, 2008;Tang et al., 2008).

However, one of the major assumptions in those publications was that all control tactics are implemented at the same time, which means that the application of pesticides can kill the released natural enemies instantly. There are three possible methods implemented in practice to avoid such antagonism when biological and chemical controls are combined: (i) spraying pesticides more frequently than releasing natural enemies; (ii) spraying pesticides less frequently than releasing Based on the above practical control methods, we first propose impulsive pest-natural enemy models with different frequencies of spraying pesticides and releasing natural enemies. The threshold conditions which guarantee that the pest dies out are derived, and several governing factors including application timing and control tactics are investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications. Experimental observations have shown that the initial densities of pest and natural enemy populations can affect classical biological control [START_REF] Burnett | Effects of initial densities and periods of infestation on the growth forms of a host and parasite population[END_REF][START_REF] Foster | Initial density of glasshouse whitefly (Trialeurodes vaporariorum (Westwood), Hemiptera) in relation to the success of suppression by Encarsia formosa Gahan (Hymenoptera) on glasshouse tomatoes[END_REF][START_REF] Jones | The effect of varying Bemisia argentifolii and Eretmocerus mundus ratios on parasitism[END_REF]. The numerical results obtained in the present paper also indicate that the simplest pest-natural enemy models with impulsive control tactics can lead to the coexistence of pests and natural enemies for a wide range of parameters, but with quite different pest amplitudes. These results indicate that the final stable states of pest-natural enemy populations depend on their initial densities as well as on their ratios, and the results are confirmed by basins of attraction of initial densities. Furthermore, the effects of random perturbations of the instant killing rates of pesticides on pests and on natural enemies, release rates and a release constant on the switch-like transitions among those attractors confirm that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial.

One of the most important questions in IPM is how many natural enemies should be released and what fraction of the pest population should be killed to avoid economic damage and reduce the pesticide applications when the pest population reaches or exceeds the ET level. In many cases, the most effective

The model with periodic integrated pest control strategies

In order to address the effects of integrated control tactics on the pest-natural enemy dynamic model, we will extend the classical Lotka-Volterra model [START_REF] Lotka | Undamped oscillations derived from the law of mass action[END_REF]Volterra, 1931) by introducing an IPM strategy such as releasing of natural enemies (Parker, 1971) or spraying pesticide at a critical time and examine the consequences of population densities changing very rapidly. For instance, impulsive reduction of the pest population is possible by trapping the pests and/or by poisoning them with chemicals. An impulsive increase of the natural enemy density can be achieved by laboratory-based breeding followed by releases into the field (Tang and Chen, 2004;Tang and Cheke, 2005;Tang et al., 2005;Tang and Cheke, 2008).

A c c e p t e d m a n u s c r i p t

The simplest case is to assume that at every period T a perturbation which incorporates a proportional (denoted by p 1 ) decrease of the insect pest and proportional (denoted by p 3 ) increase of the natural enemies and an introduction constant σ for the natural enemies which does not depend on its population sizes are considered. That is, we have the following impulsive differential equation with a fixed moment

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dx(t) dt = rx(t) [1 -δx(t)] -bx(t)y(t), dy(t) dt = y(t)(cx(t) -a), ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ t = nT, x(nT + ) = (1 -p 1 )x(nT ), y(nT + ) = (1 + p 3 )y(nT ) + σ, ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ t = nT, (2.1)
where r, δ, b, c and a are positive constants, T is the period of the impulsive effect, q 1 = 1p 1 and q 3 = 1 + p 3 with 0 < q 1 ≤ 1, q 3 ≥ 1. The assumptions in the model (2.1) without control strategies are: (i) The prey grows in a logistic way in the absence of any predation. (ii) The effect of the predation is to reduce the prey's per capita growth rate by a term proportional to the prey and predator populations. (iii) In the absence of any prey for sustenance the predator's death rate results in exponential decay. (iv) The prey's contribution to the predator's growth rate is cxy; that is, it is proportional to the available prey as well as to the size of the predator population.

The dynamical behavior and biological implications of the model (2.1) were extensively studied, and if q 3 exp(-aT ) < 1, then the complete expression of the 'pest-eradication' periodic solution of system (2.1) over the n-th time interval t 0 = nT < t ≤ (n + 1)T can be described as follows

(0, y * (t)) = 0, σ exp(-a(t -nT )) 1 -q 3 exp(-aT ) , (2.2)
which is globally asymptotically stable provided that the following threshold

condition r < 1 T ln 1 q 1 + bσ T a
1exp(-aT ) 1q 3 exp(-aT )

(2.3) holds true. Equivalently, we can define the threshold condition

R 0 = q 1 exp rT + bσ a exp(-aT ) -1 1 -q 3 exp(-aT ) (2.4)
and if R 0 < 1 then the pest eradication periodic solution is globally asymptotically stable.

In particular, let q 3 = 1 and σ = 0 (chemical control only), and the pest-eradication periodic solution (2.2) is globally stable ((0, 0) in this case) if the intrinsic growth rate of the pest population satisfies

r < 1 T ln 1 q 1
which means that if the intrinsic growth rate is less than the mean pest-killing rate due to an insecticide application over period T , then the pest population eventually goes to extinction.

Similarly, let q 1 = 1 (biological control only), then the condition which guarantees the global stability of the pest-eradication periodic solution becomes:

r < bσ T a 1 -exp(-aT ) 1 -q 3 exp(-aT )
which means that if the intrinsic growth rate is less than the mean predation rate over period T , then the pest population will become extinct eventually.

However, for an IPM strategy (q 1 > 0, q 3 ≥ 0 and σ > 0 here) inequality (2.3) indicates that if the intrinsic growth rate of the pest population is less than the summation of the mean killing rate and the mean predation rate over period T , then the pest population will tend to zero. Theoretically, this confirms that an IPM strategy is more effective than any single control strategy. The complex

Optimum timing and rates of pesticide application and release of natural enemies

To take full advantage of existing biological control agents, it is important to understand the impact of insecticides on valuable natural enemy species. In general, pesticides tend to be harmful to most natural enemies (Ruberson, 1998), which may be associated with acute toxicity. Understanding the acute toxicity of insecticides to natural enemies is important and relevant to IPM. In fact, natural enemies must search for their prey, they are generally very mobile and spend a considerable time moving across plant tissue. This increases the likelihood that they will contact the insecticide and indicates that natural enemies are generally more adversely affected by chemical insecticides than the target pest, and even worse pest populations may be induced by the application of insecticides which kill both the pest and their natural enemies [START_REF] Debach | Biological control by natural enemies[END_REF].

Undoubtedly, mathematical modelling is one of the key tools for understanding the interactions among pest, natural enemies and pesticides [START_REF] Barclay | Models for pest control using predator release, habitat management and pesticide release in combination[END_REF][START_REF] Barlow | A model for the effect of Sphecophaga vesparum as a biological control agent of the common wasp in New Zealand[END_REF]. Recently, continuous and discrete predator-prey models concerning IPM strategies have been developed and investigated [START_REF] Barclay | Models for pest control using predator release, habitat management and pesticide release in combination[END_REF][START_REF] Barclay | Predator-prey models with added mortality[END_REF][START_REF] Carpenter | Effect of control measures on pest populations subject to regulation by parasites and pathogens[END_REF][START_REF] Hassell | Insecticides in host-parasitoid interactions[END_REF][START_REF] May | Host-parasitoid systems in patchy environments: a phenomenological model[END_REF]Tang and Chen, 2004;Tang andCheke, 2005, 2008). When insecticide timing also leads to the death of parasitoids, discrete host-parasitoid models have been proposed and four different cases have been investigated according to the timing of application A c c e p t e d m a n u s c r i p t [START_REF] Beddington | Characteristics of successful natural enemies in models of biological control of insect pests[END_REF][START_REF] Hassell | Insecticides in host-parasitoid interactions[END_REF][START_REF] Hassell | Stability in insect host-parasite models[END_REF][START_REF] May | Population dynamics and biological control[END_REF]Waage and Hassell, 1982;Waage et al., 1985).

As noted in the introduction, all models (such as model (2.1)) developed before assumed that all control tactics are applied simultaneously. In the present work, let us take the simplest case where in each impulsive point τ n there is an insecticide application that kills a constant fraction (denoted by p 1 ) of the pests and which, in addition, can kill a proportion (denoted by p 2 ) of natural enemies, and in each impulsive point λ m we release a proportional amount of the natural enemies (denoted by p 3 ) and there is an introduction constant σ for the natural enemies which does not depend on the sizes of the populations. These modifications result in the following model

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dx(t) dt = rx(t) [1 -δx(t)] -bx(t)y(t), dy(t) dt = y(t)(cx(t) -a), ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ t = τ n , t = λ m , x(τ + n ) = (1 -p 1 )x(τ n ), y(τ + n ) = (1 -p 2 )y(τ n ), ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ t = τ n , y(λ + m ) = (1 + p 3 )y(λ m ) + σ, t = λ m (3.1)
where

τ n (n = 1, 2, • • •) and λ m (m = 1, 2, • • •
) are impulsive point series at which the chemical (or cultural) control tactics and biological control strategies are applied, respectively. Denote q 1 = 1p 1 , q 2 = 1p 2 and q 3 = 1 + p 3 throughout the paper with 0 < q 1 , q 2 ≤ 1, q 3 ≥ 1 and σ ≥ 0. It is thus possible to rank the different patterns of insecticide application in terms of their dynamic effects in relation to the timing of natural enemy releases. From a practical point of view, we consider several different cases as follows in terms of the timing of IPM applications.

Case 1 Pesticide applications more frequent than releases of natural enemies.

Assume λ m+1λ m ≡ T N for all m(m ∈ N ), where T N is the period of releasing

A c c e p t e d m a n u s c r i p t

natural enemies. For this case the model (3.1) is said to be a T N periodic system if there exists a positive integer k P such that

τ n+k P = τ n + T N .
This implies that in each period T N , k P pesticide applications are used.

Case 2 Natural enemy releases more frequent than pesticide applications.

Assume τ n+1τ n ≡ T P for all n(n ∈ N ), where T P is the period of pesticide applications. For this case the model (3.1) is said to be a T P periodic system if there exists a positive integer k N such that

λ m+k N = λ m + T P .
This implies that in each period T P , k N natural enemy releases are applied.

Case 3 Chemical and biological control tactics applied with different periods.

Assume λ m+1λ m ≡ T N for all m, and τ n+1τ n ≡ T P for all n. In this case, T P is the period of pesticide applications, T N is the period of natural enemy releases, m, n ∈ N . Denote ρ = T P /T N , then ρ either is rational (i.e. T P and T N are rational dependent) or is irrational (i.e. T P and T N are rational independent). If ρ is rational, then ρ = p/q, p, q ∈ N and p, q are relatively prime. Let T 0 = pT N (= qT P ), then the model (3.1) is T 0 periodic system. This means that if ρ is rational, the model (3.1) can be investigated by using similar methods as Cases 1 and 2; If ρ is irrational, then the dynamical behavior of model (3.1) becomes more complex. For this special case, it is quite difficult to investigate the dynamical behavior theoretically, please see more details from the reference [START_REF] Liu | Global dynamics of the periodic logistic system with periodic impulsive perturbations[END_REF]).

The main purposes of the following are to focus on Cases 1 and 2, and investigate the effects of timing of application of IPM tactics and rates of spraying

A c c e p t e d m a n u s c r i p t

or releasing on the pest management. We also study the effects of initial densities of pest and natural enemies on the pest control and dynamical complexities.

Dynamical analysis of Case 1 and its biological implications

For Case 1, there are k P pesticide applications during period T N . Denote

i = τ i+1 -τ i , i = 0, 1, 2, • • • , k P , where 0 = τ 1 , k P = T N -τ k P .
Firstly, the basic properties of the following subsystem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dy(t) dt = -ay(t), t = λ m y(τ + n ) = q 2 y(τ n ), t = τ n , y(λ + m ) = q 3 y(λ m ) + σ, t = λ m (4.1)
play a key role in analyzing the pest control.

It is shown in Appendix A, that there exists a globally stable periodic solution y T N (t) for the subsystem (4.1) if the inequality

q 3 q kp 2 exp(-aT N ) < 1 (4.2)
holds true. Therefore, the complete expression for the 'pest-eradication' periodic solution of system (3.1) over the h-th time interval hT N < t ≤ (h + 1)T N is given by 0, y T N (t) . Furthermore, if the following threshold condition

R N 0 = q k P 1 exp ⎡ ⎣ rT N + b a Y * k P j=0 q j 2 e -a( j i=0 i) -e -a( j-1 i=0 i) ⎤ ⎦ < 1 (4.3)
is satisfied, then the pest-eradication periodic solution 0, y T N (t) is globally attractive, where Y * = σ/ 1q 3 q k P 2 exp(-aT N ) .

A c c e p t e d m a n u s c r i p t

In particular, if i ≡ for all i = 0, 1, 2, • • • , k P , then the above condition becomes as follows:

R N 0 = q k P 1 exp rT N + b a Y * e -a -1 1 -q k P +1 2 e -(k P +1)a 1 -q 2 e -a < 1 i.e. R N 0 = q k P 1 exp rT N + b a Y * e -a -1 1 -q k P +1 2 e -aT N 1 -q 2 e -a < 1. (4.4)
Equivalently, the above inequality can become as follows

r < 1 T N ln 1 q k P 1 + b aT N Y * 1 -e -a 1 -q k P +1 2 e -aT N 1 -q 2 e -a
which can be explained as for those examples shown in the section 2.

What we want to address in the following is how control tactics including killing rates p 1 and p 2 , release rate p 3 and release constant σ, timing of pesticide application τ i (or k P ) and timing of release period T N affect the threshold condition R N 0 .

In Fig. 2(A-C), we fixed the release period T N and let the killing rate p 2 vary.

The simulation results indicate that if the pesticide kills the natural enemies with a relatively higher killing rate p 2 (for example p 2 = 0.07), the threshold value R N 0 is a monotonically increasing function with respect to the number of pesticide applications k P (Fig. 2(A)). This shows that if the pesticide has a strong effect on the natural enemies, repeated use of the same pesticides can result in target pest resurgence. If the killing rate p 2 on the natural enemies is slightly reduced from 0.07 to 0.05, Fig. 2(B) shows that the threshold value R N 0 is not monotonic with respect to the number of pesticide applications k P . So in this case we must carefully select the number of pesticide applications (two or three events in this case). If the pesticides do not kill the natural enemies so much, Fig. given releasing period, the number of application pesticide within this period and the killing rate of this pesticide on natural enemies are crucial.

Biological controls can be supported by augmentation of natural predators (and / or parasites or parasitoids), which includes the introduction of naturally occurring predators at either an inundative or inoculative level (Parker, 1971;Udayagiri et al., 2000). Fig. 3 provides the details on how different release (augmentation) rates p 3 affect the threshold value R N 0 . For the given killing rates p 1 and p 2 , release period T N , and if the number of pesticide applications is relatively small, Fig. 3 shows that slightly increasing the release rate p 3 can dramatically reduce the quantity of threshold parameter R N 0 . However, for a relatively larger number of pesticide applications, different killing rates p 1 and p 2 , a different release period T N may result in target pest resurgence, as shown in Figs. 3 and4. appear to be quite sensitive to small changes in killing rates p 1 and p 2 , release constant σ and releasing period T N . All these results clarify that the effect of pesticide timing, effectiveness of natural enemies and pesticide selectivity are crucial to pest depression and resurgence. This information may also help the field operator to decide on the optimum timing for spray applications and optimum rate for release.

Experimental observations have confirmed that the initial densities of pest and natural enemy populations can affect the outcome of classical biological control [START_REF] Foster | Initial density of glasshouse whitefly (Trialeurodes vaporariorum (Westwood), Hemiptera) in relation to the success of suppression by Encarsia formosa Gahan (Hymenoptera) on glasshouse tomatoes[END_REF][START_REF] Jones | The effect of varying Bemisia argentifolii and Eretmocerus mundus ratios on parasitism[END_REF]. The results obtained for Case 1 indicate their initial densities as well as on their ratios, and those results are confirmed by basins of attraction of initial densities (Fig. 6).

To avoid insecticide resistance, resistance strategies most often involve either mixing and applying pesticides together or alternating the use of available pesticides. But can such variations in doses and types of insecticides used and the numbers of natural enemies introduced affect the dynamics? Different numbers of natural enemies released and various dosages of pesticide applications or different pesticide applications can be mathematically expressed in terms of four parameters, q 1 , q 2 , q 3 and σ in model (3.1). That is, random perturbations due to variations in the dosages applied or releases (migration) of natural enemies can be taken into account with these four additional parameters, i.e.

q 1η = q 1 + η 1 u, q 2η = q 2 + η 2 u, q 3η = q 3 + η 3 u, σ η = σ + η 4 u and u is a random variable uniformly distributed on [-1, 1] and η i > 0 for i = 1, 2, 3, 4 to represent the intensity of noise. One view of this noise is that it represents small random events of spraying, augmentation, immigration and mortality.

In order to understand how these small random perturbations affect the final state of the pest population, we numerically studied system (3.1) with respect to the switch-like transitions among the attractors shown in Fig. 5. That is, we asked do these stable attractors switch from one attractor to another once small random perturbations have been introduced? As an example, with all other parameter values fixed as in Fig. 5, it has been shown that there are three stable attractors (2.6, 1)), the stable attractor without random perturbation is an attractor at which the pest population oscillates with a larger amplitude (see Fig. 5). When small random perturbations are introduced in one of the parameters q 1 , q 2 , q 3 , σ, numerical simulations imply that this solution can switch to another attractor with smaller amplitude at a random time (Figs. 7 and8). However, extensive numerical simulations indicate that the attractors with a smaller amplitude are robust and are not affected by these types of small random perturbations. These numerical results confirm that different doses of pesticide application and natural enemy releases can influence the dynamics of the classical pest-natural enemy system, and small random perturbations on parameters q 1 , q 2 , q 3 , σ may play key roles in insect pest control.

Dynamical analysis of Case 2 and its biological implications

For Case 2 there are k N natural enemy releases during period T P . Denote

i = λ i+1 -λ i , i = 0, 1, 2, • • • , k N , where 0 = λ 1 , k N = T P -λ k P .
It is shown in Appendix B, that there exists a globally stable periodic solution y T P (t) for the subsystem (4.1) if the inequality q 2 q k N 3 exp(-aT P ) < 1 (5.1) holds true. Therefore, the complete expression for the 'pest-eradication' periodic solution of system (3.1) over the h-th time interval hT P < t ≤ (h + 1)T P is given A c c e p t e d m a n u s c r i p t by 0, y T P (t) . Furthermore, if the following threshold condition

R P 0 = q 1 exp rT P + b a Y * (exp(-a 0 ) -1) + b a [q 3 Y * exp(-a 0 ) + σ] (exp(-a 1 ) -1) + • • • + b a q k N 3 Y * exp -a k N -1 i=0 i + σ k N -1 i=0 q i 3 exp -a k N -1 j=k N -i j • (exp(-a k N ) -1)] < 1 (5.2)
is satisfied, then the pest-eradication periodic solution 0, y T P (t) is globally attractive, where Y * is given in Appendix B. In particular, if i ≡ for all i = 0, 1, 2, • • • , k P , then the above condition becomes as follows

R P 0 = q 1 e ⎡ ⎣ rT P + b a (e -a -1) ⎛ ⎝ Y * 1-q k N +1 3 e -aT P 1-q 3 e -a +σ k N i=1 (k N +1-i)q i-1 3 e -a(i-1) ⎞ ⎠ ⎤ ⎦ < 1. (5.3)
Since the release of natural enemies in this case is more frequent than spraying pesticides, the side-effects of pesticides on the natural enemy population are largely reduced. Moreover, the threshold condition R P 0 can be significantly affected by the supplemental release of natural enemies. For example, if we fixed all parameters as those in Fig. 9

A c c e p t e d m a n u s c r i p t

We can also employ the methods provided in Case 1 to investigate the effects of all other parameters on the threshold value R P 0 . Furthermore, bifurcation analyses also indicate that multiple attractors can coexist for a wide range of parameters for this case. For example, the two attractors with quite different pest amplitudes that can coexist (see Fig. 10), and the switch-like transitions between the two attractors shown in Fig. 11. This further indicates that different dosages of pesticide applied and numbers of natural enemies released are crucial for controlling pests.

Hybrid impulsive model with Economic Threshold

As mentioned before, pesticides may kill or harm natural enemies following exposure by contact, ingestion or, less commonly, by respiration. They may also affect natural enemies indirectly by killing or contaminating their hosts or prey.

An understanding of the effectiveness of natural enemies is essential to avoid applying pesticides when biological control is adequate, as shown in Cases 1 and 2.

Probably the best method for reducing the overall negative impact of chemicals on natural enemies is to apply pesticides only when necessary. In practice, the pesticides can only be applied when the density of the pest population reaches the Economic Threshold (ET), and model (3.1) becomes

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dx(t) dt = rx(t) [1 -δx(t)] -bx(t)y(t), x(t) < ET, dy(t) dt = y(t)(cx(t) -a), t = λ m , x(t + ) = (1 -p 1 )x(t), y(t + ) = (1 -p 2 )y(t), ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ x(t) = ET, y(λ + m ) = (1 + p 3 )y(λ m ) + σ, t = λ m , (6.1)

A c c e p t e d m a n u s c r i p t

where λ m (m = 1, 2, • • •) is an impulsive point series at which the natural enemies are released. For simplification, we assume λ m+1λ m = T for all m, i.e. the periodic releasing of natural enemies is applied.

From the biological point of view, the effective testing of natural enemy release strategies and timing of insecticide applications require the use of replicated treatments in independent greenhouses (Hoddle et al., 1997a,b) and the use of experimental controls either in cages or separate greenhouses (Hoddle et al., 1997a,b). Thus, experimental methods in combination with the model approaches presented in this paper are the most cost effective method. However, the intrinsic growth rate of pests, r, predator searching efficiency, b, instant killing rates q 1 and q 2 , releasing rate q 3 and constant σ, and other factors (such as ET) can vary amongst pest and natural enemy species. How do these factors affect the control strategies? In particular, what we want to know is how do the ET and various pest-natural enemy ratios affect the control strategies.

For a given ET, we suggest that successful control strategies largely depend on the initial density and pest-natural enemy ratios. To show this, we fix all parameter values as those in Fig. 12. The results shown in Fig. 12 provide some examples of different possible cases. In Fig. 12(A) the initial densities of pest-natural enemy populations are (1.8,1.7) and the simulation result indicates that the density of the pest population never reaches the given ET, which shows that the solution initiating from (1.8,1.7) is free from pesticide applications. If we set the initial densities as (0.6,1), Fig. 12(B) indicates that the system is free from chemical control after one pesticide application. If we set the initial densities as (1.2, 0.8) or (2.1,0.5) or (2.7, 0.4), Fig. 12(C-E) indicates that the system is free from chemical control after two, four or five pesticide applications. If we further

A c c e p t e d m a n u s c r i p t

increase the pest-natural enemy ratio and set the initial densities as (1.8, 0.2), the pest outbreak frequency is significantly increased, as shown in Fig. 12(F).

We have conducted many other numerical simulations (not shown here) on model (6.1) with different initial densities and different pest-natural enemy ratios, and found that there are only three possible control cases that model (6.1) has: (a)

infinite repeats of chemical control tactics such as in periodic control strategies; (b) finite numbers of chemical control tactics; (c) no chemical control required. These results clarify that the models proposed here can help us to understand pest-natural enemy interactions, help us to design appropriate control strategies and to make management decisions in insect pest control.

Consequently, we denote the time points at which the solution reaches ET as

t n (n = 1, 2, • • •).
If mod(t n , T ) ≡ 0, a chemical control is applied at t n and after that a biological control is also applied at the same time. If mod(t n , T ) = 0, only a chemical control programme is applied. Further, denote

T n = t n -t n-1 (6.2)
with t 0 = 0 as pest-outbreak duration (or period), where n may be finite or infinite which depends on the solutions of the models.

The effect of control tactics (here parameters T, ET, q 1 , q 2 , q 3 and σ) on pest mean outbreak period (or outbreak frequency) can be calculated from the model (6.1) and formula (6.2) numerically (Fig. 13). Mean pest outbreak period is an average over several pest outbreaks (here outbreaks indicate that the densities of the pest reach the given ET) or pest cycle periods. Model (6.1) predicts that the densities of the pest do not reach the ET if the natural enemies are released frequently enough (here smaller T , Fig. 13(A)), and the mean outbreak period is decreasing as the release period T or survival rate of the pest on pesticide

A c c e p t e d m a n u s c r i p t

application q 1 increases (Fig. 13(A) and (C)). Conversely, model (6.1) predicts that the larger ET, or survival rate (here q 2 ) of the natural enemy on the pesticide application, or releasing rate (here q 3 ) and or release constant (here σ) is, the longer is the mean outbreak period (Fig. 13(B), (D), (E) and (F)). Note that the mean outbreak period can suddenly jump from a small value to a larger value at some critical points of ET, q 3 and σ, which indicates that the selection of ET and releasing methods (inundative or inoculation release) may be crucial in prolonging the pest outbreak period. Moreover, the different ET, or different values of the release rate q 3 and or different values of the release constant σ may have the same mean outbreak period (Fig. 13(B), (E) and (F)). Finally, we emphasize here that the effects of the other parameters r, b, c, d on the mean outbreak period can be investigated similarly.

Discussion

Natural enemies are usually more susceptible to the effects of pesticides than plant-feeding hosts or prey owing to their searching habits, usually less-developed enzyme-based detoxification systems and preening behavior. Usually, target pest resurgence can occur when natural enemies are destroyed. Thus, when using integrated pest management as an approach to control insect pests one must be committed to a long term strategy. If it is decided that spraying must occur, it is best to make sure that it is carried out at a time of the day or even season when there is the lowest chance of adversely affecting natural enemies. Apart from selecting pesticides that are likely to be less harmful to natural enemies and other non-target organisms such as pollinators, further selectivity can be gained by Two possible cases are investigated firstly according to the relations between spraying frequency of pesticides and release frequency of natural enemies.

Whatever releases taken place more or less frequently than the sprays, the threshold conditions which guarantee the existence and stability of the pest eradication periodic solution are provided. Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were carefully investigated. In particular, the effects of the releasing rate and releasing constant, instantaneous killing rates of pesticides for pest and natural enemy populations, releasing and spraying period on the stability of the pest-eradication periodic solution were discussed with regard to the extent of depression or resurgence resulting from pulse pesticide applications. The results imply that the modelling methods described can help in the design of appropriate control strategies and assist management decision-making.

If the integrated control methods can not completely eradicate the pest varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial for pest control. Meanwhile, the stable pest outbreak solutions with large amplitudes can switch to the stable pest outbreak solution with relatively small amplitude at a random time while the stable pest outbreak solutions with a smaller amplitude are robust to random perturbation.

These results confirm that varying dosages of pesticide applications and numbers of natural enemies released are important for pest control and pesticide resistance.

In practice, a good pest control programme should reduce pest populations to levels acceptable (here ET level) to the public rather than eradication. Our second type of model is proposed based on this ideal. To reduce the pesticide application and avoid the antagonism, we assumed that the pesticides are released only when the densities of the pest population reach the ET and periodic repeated releases of natural enemies are applied. The factors which affect the pest outbreak frequency and mean outbreak period are discussed. The simulation results indicate that the pest outbreak period or frequency largely depends on the initial densities, control tactics and given ET.

We must emphasize here that proper identification of insect pests, natural The majority of this work is based on deterministic models of the pest-natural enemy interaction with pulse perturbations and ET. This raises the interesting question of how robust the models are to various forms of environmental and demographic stochasticity, as it is well known that stochasticity can significantly affect the persistence and dynamics of populations [START_REF] Bonsall | Demographic and environmental stochasticity in predator-prey metapopulation dynamics[END_REF]Tang and Heron, 2008). The switch-like behavior in response to small random perturbations confirms that environmental stochasticity has little effect on the coexistence of attractors. However, demographic noise may affect the population dynamics more broadly, and is most influential in small populations. Stochastic fluctuations at small population sizes tend to be amplified by the dynamics to cause massive population variability, i.e. demographic stochasticity has a destabilizing effect. How do environmental and demographic stochasticity affect the rich dynamic behaviors described in this paper and influence the IPM strategy, including coexistence and the structure of the basin attractors? The introduction of stochasticity also allows us to investigate the interesting question of whether stochasticity is beneficial to IPM strategies or not, a question for future research directions.

A c c e p t e d m a n u s c r i p t

This work focused entirely on the simplest pest-natural enemy model with impulsive effects and the temporal interactions of an insect pest and its natural enemy. A number of stabilizing factors such as spatial heterogeneity, density dependent growth of the pest and functional responses of the predator will be considered in future and will be reported elsewhere.

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

which has a unique steady state

Y * = σ 1 -q 3 q k P 2 exp[-aT N ] . (A.3)
Therefore if q 3 q k P 2 exp[-aT N ] < 1, then the model (4.1) has a globally stable T N periodic solution (denoted by y T N (t)), which can be calculated as follows

y T N (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Y * exp[-a(t -hT N )], t ∈ (hT N , τ 1 + hT N ], q 2 Y * exp[-a 0 ] exp[-a(t -τ 1 -hT N )], t ∈ (τ 1 + hT N , τ 2 + hT N ],
. . .

q k P 2 Y * exp -a k P -1 i=0 i e [-a(t-τ k P -hT N )] , t ∈ (τ k P + hT N , (h + 1)T N ]. (A.4)
Furthermore, we obtain the complete expression for the 'pest-eradication' periodic solution, 0, y T N (t) , of system (3.1) over the h-th time interval

hT N < t ≤ (h + 1)T N .
It follows from system (3.1) that we have y(t) > y T N (t)for t large enough and small enough. For simplification we may assume y(t) > y T N (t)holds for all t ≥ 0. Thus we have

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dx(t) dt ≤ rx(t) -bx(t)[y T N (t) -], t = τ n , x(τ + n ) = q 1 x(τ n ), t = τ n . (A.5)
Again from the comparison theorem on impulsive differential equations we get

x(τ 1 + hT N ) ≤ x(hT N ) exp τ 1 +hT N hT N r -b(y T N (t) -) dt , x(τ 2 + hT N ) ≤ x((τ 1 + hT N ) + ) exp τ 2 +hT N τ 1 +hT N r -b(y T N (t) -) dt = q 1 x(τ 1 + hT N ) exp τ 2 +hT N τ 1 +hT N r -b(y T N (t) -) dt , . . . x((h + 1)T N ) ≤ x((τ k P + hT N ) + ) exp (h+1)T N τ k P +hT N r -b(y T N (t) -) dt = q 1 x(τ k P + hT N ) exp (h+1)T N τ k P +hT N r -b(y T N (t) -) dt ,
((h + 1)T N ) ≤ q k P 1 x(hT N ) exp τ 1 +hT N hT N r -b(y T N (t) -) dt + k P -1 j=1 τ j+1 +hT N τ j +hT N r -b(y T N (t) -) dt + (h+1)T N τ k P +hT N r -b(y T N (t) -) dt = q k P 1 x(hT N ) exp ⎡ ⎢ ⎣(r + b )T N + b a Y * ⎛ ⎜ ⎝e -a 0 -1 + q 2 ⎛ ⎜ ⎝e -a 1 i=0 i -e -a 0 ⎞ ⎟ ⎠ +q 2 2 ⎛ ⎜ ⎝e -a 2 i=0 i -e -a 1 i=0 i ⎞ ⎟ ⎠ + q 3 2 ⎛ ⎜ ⎝e -a 3 i=0 i -e -a 2 i=0 i ⎞ ⎟ ⎠ + • • • + q k P 2 ⎛ ⎜ ⎜ ⎝ e -a k P i=0 i -e -a k P -1 i=0 i ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ = q k P 1 x(hT N )e ⎡ ⎢ ⎣ (r+b )T N + b a Y * k P j=0 q j 2 ⎛ ⎜ ⎝ e -a j i=0 i -e -a j-1 i=0 i ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ = x(hT N )R N (A.7)
where -1 i=0 i = 0 and

R N = q k P 1 exp ⎡ ⎣ (r + b )T N + b a Y * k P j=0 q j 2 e -a( j i=0 i) -e -a( j-1 i=0 i) ⎤ ⎦ . Therefore, if R N 0 < 1 then x(hT ) ≤ x(0 + )(R N 0 ) h and x(hT ) → 0 as h → ∞.
Consequently, x(t) → 0 as t → ∞. Similarly, we can prove y(t) → y T N (t) as t → ∞. This indicates that if R N 0 < 1 then the pest eradication periodic solution 0, y T N (t) is globally attractive.

Appendix B

In any given time interval (hT P , (h + 1)T P ], where h is a positive integer, we investigate the dynamical behavior of model (4.1). Integrating the first equation of At time λ 1 + hT P , one release of natural enemies occurs and y((λ 1 + hT P ) + ) = q 3 y(hT + P ) exp[-aλ 1 ] + σ = q 3 y(hT

) + ) = q 3 y((λ 1 + hT P ) + ) exp[-a 1 ] + σ = q 2 3 y(hT + P ) exp[-a( 0 + 1 )] + q 3 σ exp[-a 1 ] + σ.
Similarly, we have y((λ 3 + hT P ) + ) = q 3 y((λ 2 + hT P ) + ) exp[-a 2 ] + σ = q 3 3 y(hT

+ P ) exp[-a( 0 + 1 + 2 )] +q 2 3 σ exp[-a( 1 + 2 )] + q 3 σ exp[-a 2 ] + σ = q 3 3 y(hT + P ) exp[-a( 0 + 1 + 2 )] + σ 2 i=0 q i 3 exp -a 2 j=3-i j .
By induction, we can see that

y (λ k N + hT P ) + = q k N 3 y(hT + P ) exp ⎡ ⎣ -a k N -1 i=0 i ⎤ ⎦ + σ k N -1 i=0 q i 3 exp ⎡ ⎣ -a k N -1 j=k N -i j ⎤ ⎦ and
y(t) = y (λ k N + hT P ) + exp[-a(t -λ k N -hT P )], t ∈ (λ k N + hT P , (h + 1)T P ], (B.1)
At time (h + 1)T P , one pesticide application occurs and y((h + 1)T

+ P ) = q 2 y (λ k N + hT P ) + exp[-a(T P -λ k N )] = q 2 q k N 3 y(hT + P ) exp -a k N i=0 i + q 2 σ k N -1 i=0 q i 3 exp -a k N j=k N -i j . (B.2)

A c c e p t e d m a n u s c r i p t

Denote Y h = y(hT + P ), then we have the following difference equation

Y h+1 = q 2 q k N 3 exp -a k N i=0 i Y h + q 2 σ k N -1 i=0 q i 3 exp -a k N j=k N -i j = q 2 q k N 3 exp [-aT P ] Y h + q 2 σ k N -1 i=0 q i 3 exp -a k N j=k N -i j = φY h + ψ
which has a unique steady state

Y * = ψ 1 -φ . (B.3)
Therefore if q 2 q k N 3 exp [-aT P ] < 1, then model (4.1) has a globally stable T P periodic solution (denoted by y T P (t)), which can be calculated as follows

y T P (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Y * exp[-a(t -hT P )], t ∈ (hT P , λ 1 + hT P ], [q 3 Y * exp(-a 0 ) + σ] exp[-a(t -λ 1 -hT P )], t ∈ (λ 1 + hT P , λ 2 + hT P ],
. . .

q k N 3 Y * exp -a k N -1 i=0 i + σ k N -1 i=0 q i 3 exp -a k N -1 j=k N -i j e [-a(t-λ k N -hT P ) , t ∈ (λ k N + hT P , (h + 1)T P ]. (B.4)
Once again for Case 2, we obtain the complete expression for the 'pest-eradication' periodic solution, 0, y T P (t) , of system (3.1) over the h-th time interval hT P < t ≤ (h + 1)T P .

It follows from system (3.1) that we have y(t) > y T P (t)for t large enough and small enough. For simplification we may assume y(t) > y T P (t)holds for all t ≥ 0. Thus we have

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dx(t) dt ≤ rx(t) -bx(t)[y T P (t) -], t = τ n , x(τ + n ) = q 1 x(τ n ), t = τ n .
(B.5)

A c c e p t e d m a n u s c r i p t

Again from the comparison theorem on impulsive differential equations we get

x((h + 1)T P ) ≤ x(hT + P ) exp

(h+1)T P hT P r -b(y T P (t) -) dt = x(hT + P ) exp λ 1 +hT P hT P r -b(y T P (t) -) dt + k N -1 j=1 λ j+1 +hT P λ j +hT P r -b(y T P (t) -) dt + (h+1)T P λ k N +hT P r -b(y T P (t) -) dt = x(hT + P ) exp (r + b )T P + b a Y * (exp(-a 0 ) -1) + b a [q 3 Y * exp(-a 0 ) + σ] (exp(-a 1 ) -1) + • • • + b a q k N 3 Y * exp -a k N -1 i=0 i + σ k N -1 i=0 q i 3 exp -a k N -1 j=k N -i j • (exp(-a k N ) -1)] = x(hT + P )Θ P , (B.6)
which indicates that

x((h + 1)T + P ) ≤ q 1 x(hT + P )Θ P = x(hT + P )R P (B.7)

where -1 i=0 i = 0.

Therefore, if R P 0 < 1 then x(hT + P ) ≤ x(0 + )(R P 0 ) h and x(hT + P ) → 0 as h → ∞.

Consequently, x(t) → 0 as t → ∞. Similarly, we can prove y(t) → y T P (t) as t → ∞. This indicates that if R P 0 < 1 then the pest eradication periodic solution 0, y T P (t) is globally attractive.

In particular, if i ≡ for all i = 0, 1, 2, • • • , k N , then the condition R P 0 < 1 becomes as follows: with φ = q 2 q k N 3 exp [-aT P ] and

R P 0 = q 1 exp rT P + b a e (-a ) -1 Y * + q 3 Y * e (-a ) + • • • + q k N 3 Y * e (-ak N ) +σ + • • • + σ k N -1 i=0 q i 3 exp -a k N -1 j=k N -i j = q 1 e ⎡ ⎣ rT P + b a (e -a -1) ⎛ ⎝ Y * 1-q k N +1 3 e -aT P 1-q 3 e -a +σ k N i=1 (k N +1-i)q i-1 3 e -a(i-1)
ψ = q 2 σ k N -1 i=0 q i 3 exp -a k N j=k N -i = q 2 σ k N -1 i=0 q i 3 exp [-a(i + 1) ] = q 2 σ exp(-a )-q k N 3 exp(-aT P ) 1-q 3 exp(-a )
. (A) q 1 = 0.45, q 2 = 0.94, T N = 4.2; (B) q 1 = 0.65, q 2 = 0.94, T N = 4.2; (C) q 1 = 0.45, q 2 = 0.94, T N = 4; (D) q 1 = 0.65, q 2 = 0.98, T N = 4.2. (B) q 1 = 0.65, σ = 5, T N = 14; (C) q 1 = 0.75, q 2 = 0.95, T N = 14; (D) q 1 = 0.75, q 2 = 0.95, σ = 5.

Figure 5: Three coexisting attractors of system (3.1) with parameters as follows: r = 1.5, b = 1, a = 0.6, q 1 = 0.85, q 2 = 0.95, q 3 = 4.12, σ = 0.5, δ = 0, c = 0.3, k N = 2, T N = 4. The initial conditions are: (A-B) (2.6, 2); (C-D) (2.6, 1); (E-F) (0.6, 1.5). Figure 8: Attractors' switch-like behavior of system (3.1) with small random perturbation on parameters q 1 and q 3 , i.e. η 3 = 0.1, η 1 = 0.05 and η 2 = η 4 = 0. The other parameters are identical to those in Fig. 5. 0.2, T P = 5, q 1 = 0.75, q 2 = 0.9, q 3 = 1.1 with i ≡ .

Figure 10: Two coexisting attractors of system (3.1) with parameters as follows: r = 3, b = 1, a = 0.2, q 1 = 0.65, q 2 = 0.98, q 3 = 1.2, σ = 0.5, δ = 0, c = 0.3, T P = 8.22, k N = 2. The initial conditions are: (A-B) (0.6, 1); (C-D) (0.6, 0.7).

A c c e p t e d m a n u s c r i p t

Figure 11: Attractors' switch-like behavior of system (3.1) with small random perturbation on parameters q 1 and q 3 , i.e. η 3 = 0.1, η 1 = 0.05 and η 2 = η 4 = 0. The other parameters are identical to those in Fig. 10.

Figure 12: Illustrating the effects of ET and initial densities of pest and natural enemy populations of model (6.1) on the control strategies. The parameters are fixed as follows: r = 3, b = 1, a = 0.3, q 1 = 0.65, q 2 = 0.95, q 3 = 1.2, σ = 1, δ = 0, c = 0.3, T = 4. Initial densities in (A) is (1.8, 1.7), in (B) is (0.6,1), in (C) is (1.2, 0.8), in (D) is (2.1,0.5), in (E) is (2.7, 0.4), and in (F) is (1.8, 0.2).

Figure 13: The mean outbreak period of model (6.1), as a function of T in (A), as a function of ET in (B), as a function of q 1 in (C), as a function of q 2 in (D), as a function of q 3 in (E), and as a function of σ in (F). The basic parameter set is fixed as follows: r = 2, b = 1, a = 0.6, q 1 = 0.65, q 2 = 0.95, q 3 = 1.2, σ = 1, δ = 0, c = 0.1, T = 4, ET = 5. Initial densities are (H 0 , P 0 ) = (1, 1). Simulations were run for 100 × T to rule out transients, and then run for 50 × T to get T n which allows us determine the average outbreak period.

  and (iii) releasing natural enemies frequently and spraying pesticides only when pest densities reach the ET.

  spraying rate has not been identified as it will vary depending on crop type and target host density. To avoid antagonism and reduce the pesticide applications, the pesticide is sprayed only when it is necessary, i.e. when the pest population density reaches the ET. With this in mind we developed a new hybrid impulsive dynamical model to involve a periodic releasing of natural enemies, while chemical control is applied only when the densities of the pest population reach the given ET. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics. The results also show that the selection of ET, selection of the pesticide and selection of the releasing methods (inundative or inoculation release) may be crucial in prolonging the pest outbreak period.

  model (2.1) has also been addressed by using numerical investigations(Tang and Cheke, 2005; Tang et al., 2005).

  2(C) clarifies that the threshold value R N 0 is a monotonically decreasing function with respect to A c c e p t e d m a n u s c r i p t the number of pesticide applications k P . All these simulations show that for a

Fig. 4

 4 Fig.4 shows the effects of different parameter sets on the quantity of the threshold condition R N 0 . All simulation results shown in Fig.4 indicate that the R N 0

  Fig.5. The final stable states of pest and natural enemy populations depend on

  . If we choose the initial condition (x 0 , y 0 ) = (0.6, 1.5) (or

  and chose different releasing constant σ and different k N , the simulation results indicate that slight increases of the release constant σ can significantly reduce the threshold value R P 0 (Fig.9), while increasing the number of natural enemy releases as well. This clarifies that repeated releases of a small number of natural enemies at a critical time of the season can successfully suppress the pest population. In practice, an example of inoculative release occurs in greenhouse production of several crops. Periodic releases of the parasitoid Encarsia formosa are used to control greenhouse whitefly Trialeurodes vaporariorum, and the predaceous mite, Phytoseiulus persimilis, is used for control of the two-spotted spider mite Tetranychus vrticae (Parker, 1971).

  involving minimization of the pesticide concentration and appropriate timing of pesticide applications.Due to the antagonism between chemical and biological methods, understanding the range of possible ecological interactions among pest, natural enemy and pesticides is quite important for successful pest control. Here we have extended the classical continuous pest-natural enemy models to include an IPM control strategy in order to (a) understand why different release methods result in different pest control outcomes; (b) to estimate and predict the timing of a pesticide application and the instantaneous killing rates of insecticide applications on pest and natural enemy, and (c) to investigate the effects of initial densities of pest-natural enemy populations on insect pest control.

  .e. the threshold condition does not hold true any more, the pest population can have outbreaks at different scales. By extensive numerical bifurcation investigations, we found that when choosing different parameter spaces, multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters. The results indicate that the dynamic behavior of a population may be affected dramatically by small changes in the values of initial densities of the pest and natural enemy populations, and the switch-like transitions among these attractors showing that

  basic knowledge of economic thresholds are essential for an IPM programme to be successful, which can help us to determine parameters p 1 , p 2 , p 3 and ET. Therefore, in order to keep track of both pest and natural enemy populations, regular field monitoring are necessary, and these data are quite important for making management decisions. However, regular field monitoring needs lots of manpower and time, and consequently increases the costs. Further, the ET is dynamic and depends on many factors, see more details on how to determine the ET from the literature (Tang and Cheke, 2005).

  from hT P to λ 1 + hT P yields y(t) = y(hT + P ) exp[-a(t -hT P )], t ∈ (hT P , λ 1 + hT P ].

Figure 1 :

 1 Figure 1: Economic Injury Level (EIL) = lowest population density that will cause economic damage. Economic Threshold (ET) = population density at which control measures should be introduced to prevent an increasing pest population from reaching the economic injury level. The arrows indicate points when pest levels exceed the economic threshold and an IPM strategy would be applied.

Figure 2 :

 2 Figure 2: The effects of number of pesticide applications and release period T N on the threshold level R N 0 . The parameter values are as follows: r = 1.1755, b = 1, a = 0.307, q 1 = 0.85, q 3 = 1.0, σ = 5.1, and k p = 1, 2, 3, • • • , 12 with i ≡ .

Figure 3 :

 3 Figure 3: The effects of the number of pesticide applications and release rate q 3 on the threshold level R N 0 . The parameter values are as follows: r = 1.3, b = 1, a = 0.42, σ = 0.5, and k p = 1, 2, 3, • • • , 12 with i ≡ . (A) q 1 = 0.45, q 2 = 0.94, T N =

Figure 4 :

 4 Figure 4: The effects of times of spraying pesticides and parameter sets on the threshold level R N 0 . The parameter values are as follows: r = 1.2, b = 1, a = 0.307, q 3 = 1.0, and k p = 1, 2, 3, • • • , 12 with i ≡ . (A) q 2 = 0.95, σ = 5, T N = 14;

Figure 6 :

 6 Figure 6: Basins of attraction of the three attractors shown in Fig.5 with the parameters identical to those in Figure 5. The white, green and red points are attracted to the attractors shown in Fig.5 from top to bottom, respectively.

Figure 7 :

 7 Figure 7: Attractors' switch-like behavior of system (3.1) with small random perturbation on parameters q 2 and σ, i.e. η 2 = 0.05, η 4 = 0.05 and η 1 = η 3 = 0. The other parameters are identical to those in Fig.5.

Figure 9 :

 9 Figure 9: The effects of times of releasing natural enemies and releasing constant σ on the threshold level R N 0 . The parameter values are as follows: r = 0.8, b = 1, a =

  + P ) exp[-a 0 ] + σ.Again, integrating the first equation of model (4.1) from λ 1 + hT P to λ 2 + hT P yieldsy(t) = y((λ 1 + hT P ) + ) exp[-a(tλ 1 -hT P )], t ∈ (λ 1 + hT P , λ 2 + hT P ].At time λ 2 + hT P , one release of natural enemies occurs and y((λ 2 + hT P

  Neuenschwander, P.,Herren, H.R., 1988. Biological Control of the Cassava Mealybug, Phenacoccus manihoti, by the Exotic Parasitoid Epidinocarsis lopezi in Africa. Phil. Trans. R. Soc. Lond. B 318,319-333. Udayagiri, S., Norton, A.P., Welter, S.C., 2000. Integrating pesticide effects with inundative biological control: interpretation of pesticide toxicity curves for Anaphes iole in strawberries. Entomologia Experimentalis et Applicata 95,87-95.

	Parker, F.D., 1971. Management of pest populations by manipulating densities of
	both host and parasites through periodic releases, In: Huffaker, C. B., ed.
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Acknowledgements

We would like to thank the referees for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper. The first author is supported by the National Natural Science Foundation of China NSFC 10871122(S.Y. Tang).

Appendix A

In any given time interval (hT N , (h + 1)T N ], where h is a positive integer, we investigate the dynamical behavior of model (4.1). In fact, integrating the first equation of model (4.1) from hT N to τ 1 + hT N yields

At time τ 1 + hT N , one pesticide application occurs and y((τ 1 + hT N ) + ) = q 2 y(hT + N ) exp[-aτ 1 ] = q 2 y(hT

Again, integrating the first equation of model (4.1) from τ 1 + hT N to τ 2 + hT N yields

At time τ 2 + hT N , one time pesticide application occurs and

By induction, we can see that

for all t ∈ (τ k P + hT N , (h + 1)T N ]. At time (h + 1)T N , release of natural enemies occurs once and

Denote Y h = y(hT + N ), then we have the following difference equation q 3 =4 q 3 =4.5 q 3 =5 q 3 =5.5 q 3 =4 q 3 =4.5 q 3 =5 q 3 =5.5 q 3 =4 q 3 =4.5 q 3 =5 q 3 =5.5


Dr Raashda A Sulaiman
email: raashda.sulaiman@rwh-tr.nhs.uk

Mourad H Labib

Keywords: Impaired fasting glycaemia, impaired glucose tolerance, oral glucose tolerance test, diabetes

Impaired fasting glycaemia (IFG) is an indication for oral glucose tolerance test (OGTT). World health organisation (WHO) and International Diabetes Federation (IDF) define IFG as fasting plasma glucose levels of 6.1to 6.9 mmol/L. However American Diabetes Association (ADA) still recommends a range of 5.6 to 6.9 mmol/L as IFG. We performed an audit to assess the outcome of oral glucose tolerance test (OGTT) at various cut offs of fasting plasma glucose (FPG) levels in patients at high risk of developing diabetes.

Methods:

Laboratory data on OGTT performed over a period of one year in a district general hospital, was collected. Patients with fasting plasma glucose (FPG) levels between 5.6 to 6.9 mmol/L were selected and the outcome was analysed.

Results:

Our audit shows that in patients with FPG levels of 5.6 to 6.0 mmol/L, 19% had diabetes and 43% had impaired glucose tolerance (IGT).

Conclusion:

The percentage of subjects with abnormal OGTT in our study is much higher than that of Decode study 1 (7% diabetes and 29% IGT). However Decode study had included general population while our data was collected from subjects who are at high risk of developing diabetes. We conclude that in these subjects the lower cut off level of 5.6 mmol/L for FPG should be used as an indication for OGTT.

INTRODUCTION

Type 2 diabetes is one of the major challenges facing health care services in UK. Currently there are 2.5 million people diagnosed with diabetes while more than half a million still remain undiagnosed. It has been estimated that there will be over four million people with diabetes in UK by 2025. [START_REF]Diabetes in the UK 2009-Key statistics on diabetes-NHS Evidence on diabetes[END_REF] Epidemiological evidence suggests that diabetes related complications begin in the pre diabetes state, which refers to IFG and/or IGT. Increasing insulin resistance and progressive beta cell failure result in rising glycaemia in the non diabetic range, which could eventually progress to diabetes. [START_REF] Nichols | Progression from newly acquired impaired fasting glycaemia to type 2 diabetes[END_REF] Currently, two definitions for IFG are in use. The WHO and recently the IDF agreed for the range of IFG as a FPG of 6.1 to 6.9 mmol/L, in venous plasma sample using laboratory-based methods. [START_REF]report of a WHO/IDF consultation[END_REF] However, ADA still recommends a range of 5.6 to 6.9 mmol/L as IFG [Table1]. [START_REF] Nichols | Progression from newly acquired impaired fasting glycaemia to type 2 diabetes[END_REF] FPG (mmol/L) 2-hr plasma glucose in OGTT (mmol/L) IFG WHO/IDF: 6.1-6.9 ADA : 5.6 -6.9 < 7.8 IGT <7.0 7.8 -11.0 We audited the outcome of OGTT performed over a period of one year in patients with FPG concentrations of 5.6 to 6.9 mmol/L to assess the frequency of abnormal glucose tolerance between the two guidelines. Requests from the antenatal clinic were excluded. The 75 g OGTT was performed following an overnight fast. Plasma glucose was measured using a glucose oxidase method on Vitros 950 analyser (Ortho-Clinical Diagnostics, Inc.).The inter-assay coefficient of variation (CV) for glucose was 1.5% and 1.1% at 5.0 mmol/L and 16.0 mmol/L respectively.

We selected patients with FPG concentrations between 5.6 to 6.9 mmol/L and analysed the outcome of the OGTT, as defined by WHO criteria (normal: 2hr plasma glucose <7.8 mmol/L; IGT: 2hr plasma glucose ≥7.8 to11.0 mmol/L; and diabetes: 2hr plasma glucose ≥11.1 mmol/L). Patients were then categorised into three groups, one with FPG 5.6 to 6.9 mmol/L, according to ADA criteria for IFG, and second group with FPG 6.1 to 6.9 mmol/L as in WHO criteria for IFG and a third group with lower cut point of 5.6 to 6.0 mmol/L (the difference between the two criteria). We compared the incidence of IGT and diabetes in these three groups. Data was analysed using The ADA criteria for IFG with a lower cut point of 5.6 mmol/L significantly increases the number of people who have IFG but not IGT/DM. On the other hand, the percentage of people with both IFG and IGT increases with the WHO criteria for IFG with a higher cut point of 6.1 mmol/L.

Several studies have reported that 6.1 mmol/L cut point for FPG, more strongly predicts future diabetes and cardiovascular mortality than 5.6 mmol/L cut point. 5,6 There is not enough data on clinical outcomes of intervention in people with IFG as compared to IGT. It has been argued that for a significantly larger number of people with ADA-defined IFG, who apparently have much lower incidence of IGT, preventive strategies are not cost-effective as there is not enough evidence of benefit. [START_REF]report of a WHO/IDF consultation[END_REF] Therefore, WHO and lately the IDF have recommended that FPG cut point for IFG should remain at 6.1 mmol/L. However, our audit showed that 62% of patients with FPG levels of 5.6 -6.0 mmol/L had abnormal OGTT results. Moreover the incidence of IGT was not significantly different in both lower and higher cut off groups. Interestingly our data shows that 19% of patients with FPG of 5.6-6.0 mmol/L have diabetes.

This was surprising, as the DECODE study [START_REF] Labib | Is fasting glucose sufficient to define diabetes? Epidemiological data from 20 European studies[END_REF] which assessed the data from 20

European trials found that using WHO criteria for IFG (6.1 -6.9 mmol/L), only 29% had IGT and 7% had diabetes. One explanation for the difference could be that the DECODE study had included the general population while our data was collected from patients who had OGTT performed because of additional risk factors. For example, 18% of our subjects were referred from the weight management clinic and also the mean age of our subjects (60 yrs) was higher than that of Decode population (54yrs).

The proportion of individuals with newly diagnosed type 2 diabetes who were obese, has increased from 46% in 1996 to 56% in 2005 7 and there is a growing evidence that type 2 diabetes can be prevented or delayed by life style changes. 8,9 In Diabetes Prevention Program, the incidence of type 2 diabetes was reduced by 58% in people with IGT by life style modification. 9 It is therefore important to identify those who are at high risk for developing diabetes, so that a diabetes prevention strategy can be initiated early. We have observed in our weight management clinic that patients with IGT, are generally very motivated to make healthy changes in their lifestyle to prevent or delay diabetes. Our audit suggests that in individuals at high risk of developing diabetes, the ADA criteria with a lower cut off level of 5.6 mmol/L for FPG would be more appropriate as an indication to proceed with OGTT.

1

 1 Is using WHO criteria for impaired fasting glycaemia appropriate as an indication for OGTT in patients at high risk of developing diabetes?Is using WHO criteria for impaired fasting glycaemia appropriate as an indication for OGTT in patients at high risk of developing diabetes?

Figure 1 :

 1 Figure 1: The outcome of OGTT in patients with various levels of cutoffs for FPG

  , N.G., Luan, J., Hennings, S. and Wareham, N.J., Incidence of Type 2 Diabetes in England and its association with baseline impaired fasting glucose: The ELY study 1990-2000 Diabet Med 2007;24:200-207 6. Rijkelijkhuizen JM, Nijpels G et al : High risk of cardiovascular mortality in individuals with impaired fasting glucose is explained by conversion to diabetes: the Hoorn study. Diabetes Care 2007,30:332-336 7. Gonzalez ELM, Johansson S, Wallender MA, Rodriguez LAG:Trends in the prevalence and incidence of diabetes in the UK:1996-2005 J Epidemeol Community Health 2009,63:332-336 8. Kulzer B, Hermaan N, Schwarz P, Daniela G: Prevention of Diabetes Self Management Program (PREDIAS): Effects on weight, Metabolic Risk factors and Behavioral Outcome. Diabetes Care2009;32:1143-46 9. Diabetes Prevention Program Research Group. Reduction in the incidence of type 2 diabetes with life style intervention or Metformin. NEJM 2002;346:

Table 1 :

 1 WHO and ADA criteria of IFG and IGT

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	4

Page 4 of 12 International Journal of Clinical Practice International Journal of Clinical Practice

  We collected laboratory data on OGTT performed between 1 st January 2008 and 31 st December 2008, at Russells Hall Hospital, Dudley which is a district general hospital serving approximately 350,000 predominantly White Caucasian population. This included requests from General Practitioners and hospital outpatients on patients who had risk factors for developing diabetes such as obesity, metabolic syndrome, and a family history of diabetes.

	METHODS:
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	5

Table 2 :

 2 The outcome of OGTT in patients with various levels of cutoffs for FPG:

	GraphPad Instat version 3.00 for Windows 95 (GraphPad Software, San
	Diego, USA).

Acknowledgement:

Competing interests: None Funding : No funding was recieved for this report Guarantor : Mourad Labib

Author Contributions:

1. Raashda Sulaiman collected, analysed the data, wrote manuscript, researched literature and contributed to discussion.

2. Mourad Labib contributed to discussion, reviewed and edited manuscript.


Wojciech Leppert¹

Mikolaj Majkowicz²

Wojciech Leppert
email: wojciechleppert@wp.pl

The impact of tramadol and dihydrocodeine treatment on quality of life of patients with cancer pain

Keywords: cancer pain, dihydrocodeine, performance status, quality of life, tramadol

Background: Tramadol and dihydrocodeine (DHC) are analgesics of step 2 WHO analgesic ladder (opioids for mild to moderate pain, weak opioids) frequently used in the treatment of cancer pain of moderate intensity. The aim of the study was to assess the impact of tramadol and DHC treatment on quality of life (QL) and performance status (PS) of patients with cancer pain.

Patients and methods: Randomized, cross-over, clinical study of 40 opioid-naive patients with nociceptive cancer pain who received tramadol or DHC controlled release tablets for 7 days, then drugs switched and administered for another 7 days. Pain was assessed by VAS (visual anlogue scale), QL by EORTC QLQ C 30; PS by ECOG and Karnofsky.

Results:

From 40 patients recruited thirty completed the study. DHC treatment provided better analgesia (VAS). In QL functional scales better emotional functioning in tramadol group; better global QL and cognitive functioning in DHC group were observed. In symptom scales less fatigue, pain and sleep disturbances, less nausea and vomiting and better appetite in DHC group were noted. In tramadol group less constipation and less financial problems were observed. No differences in dyspnoea and diarrhoea were noted. ECOG and Karnofsky PS were low and did not differ between tramadol and DHC groups.

Conclusions: DHC treatment was associated with better global QL, cognitive functioning, analgesia and appetite, less fatigue, sleep disturbances, nausea and vomiting. Tramadol therapy was connected with better emotional functioning, less constipation and financial problems. PS deteriorated in both tramadol and DHC groups.

What is already known on this topic?

Tramadol and dihydrocodeine are weak opioids used in the treatment of chronic and cancer pain of moderate intensity. Analgesic efficacy and adverse effects of both opioids were assessed in a number of clinical trials and in experimental studies. However equianalgesic doses of the drugs were not clearly established in clinical trials. The most common adverse effects of tramadol is nausea, sweating and diziness and of dihydrocdeine constipation, nausea and sedation.

What does this article add?

No studies in patients with cancer pain directly compared the drugs. This is the first clinical study that addresses analgesia, adverse effects and quality of life in patients with cancer pain treated with tramadol and dihydrocodeine in a cross-over design. Equianalgesic doses of tramadol and dihydrocdeine were established, the most common adverse effects of the analgesics identified and the guidelines of prophylaxis of tramadol and dihydrocodeine adverse effects were discussed

Introduction

Tramadol and dihydrocodeine (DHC) are analgesics of step 2 of the WHO analgesic ladder (opioids for mild to moderate pain, weak opioids), which are still frequently used in the treatment of cancer pain of moderate intensity. A number of studies indicate usefulness of weak opioids especially tramadol [START_REF] Hennies | Receptor binding, analgesic and antitussive potency of tramadol and other selected opioids[END_REF][START_REF] Preston | Abuse potential and pharmacological comparison of tramadol and morphine[END_REF][START_REF] Stamer | Impact of CYP2D6 genotype on postoperative tramadol analgesia[END_REF] and DHC [START_REF] Schmidt | Affinities of Dihydrocodeine and its Metabolites to Opioid Receptors[END_REF][START_REF] Wilder-Smith | The visceral and somatic antinociceptive effects of dihydrocodeine and its metabolite, dihydromorphine. A cross-over study with extensive and quinidine-induced poor matabolizers[END_REF][START_REF] Wilder-Smith | Treatment of severe pain from osteoarthritis with slow-release tramadol or dihydrocodeine in combination with NSAID's: a randomized study comparing analgesia, antinociception and gastrointestinal effects[END_REF][START_REF] Dobrogowski | Kliniczna ocena dzialania DHC Continus u chorych z bolem przewlekłym[END_REF][START_REF] Leppert | Ocena analgezji i objawow niepozadanych tramadolu i dihydrokodeiny o kontrolowanym uwalnianiu u chorych z bolem nowotworowym -na podstawie zmodyfikowanej skali ESAS[END_REF]. A significant percentage of patients with cancer and chronic non malignant pain of moderate intensity may benefit from these drugs. However weak opioids are often neglected as a consequence of high efficacy of opioids for moderate to severe pain (opioids of step 3 WHO analgesic ladder, strong opioids) supported by clinical studies [START_REF] Mercadante | Low Morphine Doses in Opioid-Naive Cancer Patients with Pain[END_REF][START_REF] Maltoni | A validation study of the WHO analgesic ladder: a two-step vs three-step strategy[END_REF][START_REF] Marinangeli | Use of Strong Opioids in Advanced Cancer Pain: A Randomized Trial[END_REF][START_REF] Porta -Sales | WHO analgesic ladder -or lift?[END_REF]. These studies are not quite convincing because of limited number of patients recruited. It seems also appropriate to administer weak opioids in moderate pain and strong opioids in patients with severe pain intensity. Studies comparing analgesia and adverse effects of high tramadol with low morphine doses indicate better global QL [START_REF] Leppert | Ocena jakosci zycia u chorych z bolem nowotworowym leczonych tramadolem i morfina[END_REF] and less adverse effects during tramadol treatment with similar analgesia [START_REF] Grond | High -Dose Tramadol in Comparison to Low -Dose Morphine for Cancer Pain[END_REF][START_REF] Leppert | Analgesic efficacy and side effects of oral tramadol and morphine administered orally in the treatment of cancer pain[END_REF]. Analgesic effect of weak opioids may be also improved by concomitant administration of NSAIDs, metamizole or paracetamol with tramadol, codeine or DHC [START_REF] Leppert | Rola analgetykow opioidowych II stopnia drabiny analgetycznej Swiatowej Organizacji Zdrowia w leczeniu bolu nowotworowego[END_REF].

Tramadol is an opioid agonist, which apart from opioid action activates pain inhibitory system (increase of noradrenaline and serotonin level). In comparison to other opioids (morphine, pethidine, buprenorphine) tramadol displays small affinity to mu opioid receptors, rarely causes respiratory and circulatory depression and physical dependence [START_REF] Hennies | Receptor binding, analgesic and antitussive potency of tramadol and other selected opioids[END_REF]. Tramadol in the dose 100 mg causes similar analgesia to 10 mg of morphine when the drugs are given orally [START_REF] Preston | Abuse potential and pharmacological comparison of tramadol and morphine[END_REF]. The active metabolite Odesmethyltramadol possesses significant mu opioid receptor affinity and displays analgesic activity [START_REF] Stamer | Impact of CYP2D6 genotype on postoperative tramadol analgesia[END_REF]. Tramadol is available in immediate and controlled release (CR) oral formulations. Dihydrocodeine (DHC) is a semi synthetic codeine derivative, which was formed by the hydrogenation of the double tie in the main chain of the codeine molecule. DHC exerts its analgesic action through affinity to mu, kappa and delta opioid receptors. After oral administration, 60 mg DHC analgesic activity is similar to 10 mg of morphine [START_REF] Schmidt | Affinities of Dihydrocodeine and its Metabolites to Opioid Receptors[END_REF]. DHC and its metabolites: dihydromorphine, nordihydrocodeine and dihydrocodeine-6glucuronide display analgesic activity mostly through mu opioid receptors activation [START_REF] Wilder-Smith | The visceral and somatic antinociceptive effects of dihydrocodeine and its metabolite, dihydromorphine. A cross-over study with extensive and quinidine-induced poor matabolizers[END_REF]. DHC is available in CR tablets for oral administration. To our knowledge up to date no clinical studies in patients with cancer pain compared these analgesics. Quality of life (QL) assessment is increasingly important issue in patients with cancer pain. The aim of the

Patients and methods

A randomized, cross-over, clinical study of patients treated either with CR tramadol (15 patients) or CR DHC tablets (15 patients) for 7 days; after this period drugs switched for another 7 days of treatment. All patients were treated at one in-patient palliative medicine unit; patients were recruited within the period of January 2007 -June 2008. To enter the study patients had to fulfill all inclusion criteria: cancer diagnosis, age over 18, opioid-naive, no history of drugs abuse, oral route of drug administration, ability of normal communication and filling questionnaires, nociceptive baseline pain intensity (VAS: 0 -no pain, 100 -the worst possible pain) over 40 during non-opioids (NSAIDs, paracetamol, metamizol) therapy, no renal impairment, if woman not pregnant and no lactation. Visceral pain was diagnosed in 17 patients, somatic in 13: in 9 bone, in 4 somatic from soft tissues and from the skin. Primary tumor localization was as follows: lung (4 patients), colon (4), stomach (3) and pallatinal tonsil [START_REF] Preston | Abuse potential and pharmacological comparison of tramadol and morphine[END_REF]; in one patient pharynx, esophagus, gall bladder, pancreas, thyroid and suprarenal glands, kidney, prostate, breast, skin, skin melanoma, myelodysplastic syndrome, Hodgkin disease, ovary, abdominal and pelvic tumors and bone metastases from unknown primary site.

Equianalgesic dose ratio of tramadol to DHC was 5:3 when drugs were switched after 7 days of the treatment (table 1) [START_REF] Wilder-Smith | Treatment of severe pain from osteoarthritis with slow-release tramadol or dihydrocodeine in combination with NSAID's: a randomized study comparing analgesia, antinociception and gastrointestinal effects[END_REF][START_REF] Dobrogowski | Kliniczna ocena dzialania DHC Continus u chorych z bolem przewlekłym[END_REF]. The starting dose of CR tramadol was 100 mg b.i.d., CR DHC 60 mg b.i.d. titrated according to the scheme (tab. 1) to satisfactory analgesia defined as pain intensity less than 40 or decrease by at least 20 (VAS). The maximal daily doses were for tramadol 600 mg and for DHC 360 mg. In case satisfactory analgesia was not achieved the treatment was stopped, patients were switched to strong opioids. Previous treatment with NSAIDs, paracetamol or metamizol was allowed. Patients did not receive prophylactic antiemetics and laxatives when starting treatment with tramdol or DHC. EORTC QLQ C 30 and performance status (PS ECOG and Karnofsky) were assessed at baseline, at the 7th day (before drug switch) and on the 14th day of therapy. At study completion patients were asked about the preferred analgesic. All patients provided written, informed consent prior to the study procedures. The trial was accepted by the Regional Bioethics Committee at the Poznan University of Medical Sciences and it was performed according to the Declaration of Helsinki. The data were statistically analysed with a licensed statistical package (Statistica PL, version 8.0®). Pain intensity in both groups was assessed by VAS (mean and standard deviation); the t-test for paired data was used to compare analgesic effects. In descriptive statistics of functional and symptom scales of the EORTC QLQ C 30 (data not shown), ECOG and Karnofsky PS means and standard error were provided. Two-way ANOVA was used for the EORTC QLQ C 30, ECOG and Karnofsky PS results analysis. PS (ECOG and Karnofsky) results during treatment were compared to baseline by the the least significant difference (LSD) test. In all analyses p-value 0.05 was considered significant.

Results

At the entry of the trial 40 patients were recruited; thirty completed the study. From 10 patients who dropped out (data not included into results) four (1 from tramadol and three from DHC group) deteriorated during the first 7 days and could not received drugs orally, four patients treated with tramadol (tramadol/DHC group) and 2 treated with DHC (DHC/tramadol group) did not achieve satisfactory analgesia during titration (fig. 1). The mean age of patients was 70.47 ± 8.97; there were 19 women and 11 men. Results regarding adverse effects based on the modified ESAS (Edmonton Symptom Assessment System with two additional scales for vomiting and constipation) were reported in another paper [START_REF] Leppert | Ocena analgezji i objawow niepozadanych tramadolu i dihydrokodeiny o kontrolowanym uwalnianiu u chorych z bolem nowotworowym -na podstawie zmodyfikowanej skali ESAS[END_REF]. No serious adverse events were observed with tramadol and with DHC treatment. The daily tramadol doses were 286.67 ± 157.35 mg and 256.20 ± 109.33 mg; for DHC 138.87 ± 40.77 mg and 172.53 ± 95.19 mg on the 7 th and on the 14 th day of the treatment respectively. The mean pain intensity at baseline was 63.07 ± 13.48 for tramadol and 57.73 ± 8.40 for DHC (p > 0.2). During all but two days DHC analgesic effect was superior to tramadol (tab. 2). More patients in the tramadol group [START_REF] Porta -Sales | WHO analgesic ladder -or lift?[END_REF] than in the DHC group (8 patients) used rescue analgesics. There were 19 patients who preferred DHC treatment and 4 patients in favour of tramadol therapy; 7 patients assessed both analgesics as equally effective.

ANOVA results for functional scales of EORTC QLQ C 30 are shown (tab. 3). No drug effect was observed. There was no treatment time effect in physical, role (work), cognitive, and social functioning. However treatment time effect was observed in emotional functioning (p < 0.002) and global QL (p < 0.001). There was no drug and treatment time interaction effect in physical, role (work) and social functioning but significant effect in cognitive functioning (p = 0.05), emotional functioning (p < 0.013) and global QL (p < 0.006). Higher scores of emotional functioning in tramadol group, higher global QL and better cognitive ANOVA results for symptom scales of EORTC QLQ C 30 are shown (tab. 4). No drug effect was found, although in case of nausea and vomiting (p = 0.072) and sleep (p = 0.06) p-values were near significance. Treatment time effect was significant for fatigue (p < 0.036), pain (p < 0.001), sleep (p < 0.001) and finances (p < 0.023) with no effect in nausea and vomiting, dyspnoea, appetite, constipation and diarrhea. Interaction of drug and treatment time effect was observed for fatigue (p < 0.01), nausea and vomiting (p < 0.001), pain (p < 0.001), sleep (p < 0.005), appetite (p < 0.039) and constipation (p < 0.001) with no effect in dyspnoea, diarrhea and finances. Less intense fatigue, pain and sleep disturbances, nausea and vomiting and better appetite in DHC group were noted. In tramadol group less constipation and financial problems were observed. No differences in dyspnoea and diarrhoea were found.

ECOG PS of all patients deteriorated comparing baseline with day 7 (p = 0.008) and with day 14 (p = 0.001) (tab. 5). ANOVA revealed treatment time effect (p < 0.001) with no drug and no drug and treatment time interaction effect (tab. 6). No differences were found in ECOG PS between tramadol and DHC groups at baseline (p > 0.2), day 7 (p > 0.6) and day 14 (p > 0.27; LSD test). Karnofsky PS of all patients deteriorated comparing baseline with day 14 (p = 0.001). A trend was observed between baseline and day 7 (p = 0.062) (tab. 5). ANOVA revealed treatment time effect (p < 0.001) and a trend (p = 0.087) of drug and treatment time interaction with no drug influence (tab. 6). No differences were found in Karnofsky PS between tramadol and DHC groups at baseline (p > 0.61), day 7 (p > 0.92) and day 14 (p > 0.61; LSD test).

Discussion

To our knowledge no studies were performed to date comparing tramadol with DHC in patients with cancer pain. We observed better analgesic effect of DHC. A superior DHC analgesia was demonstrated in VAS, less consumption of rescue analgesics in DHC group and more patients who preferred DHC therapy. The study demonstrated possibility of tramadol to DHC and opposite way rotation, which in clinical practice may be indicated in case of adverse effects or insufficient analgesia during tramadol or DHC therapy using tramadol to DHC ratio for the oral route 5:3. A significant increase in daily tramadol doses (40%) and DHC doses (20%) in the first week of the treatment was observed. In the second week daily doses were more stable taking into account the former analgesic dose at the 7 th day (before switch) and the ratio used for drug rotation.

Tramadol and DHC were compared in few studies. Hummel et al. in a double-blind, cross-over, placebo-controlled experimental study demonstrated both drugs to be more effective when administered in the evening than in the morning; no equianalgesic tramadol/DHC dose was established [START_REF] Hummel | Analgesic Effects of Dihydrocodeine and Tramadol when Administered Either in the Morning or Evening[END_REF]. Wilder -Smith et al. compared CR tramadol with CR DHC in osteoarthritis pain not responding to NSAIDs. Dose calculation was the same as in our study (CR tramadol 100 mg b.i.d. corresponded to CR DHC 60 mg b.i.d.). Analgesia was similar but tramadol caused more adverse effects and less constipation. This trial did not comprised QL evaluation and had a parallel design. Mean daily doses of analgesics were stable over the study period (209 mg and 203 mg for CR tramadol; 129 mg and 130 mg for CR DHC at the 1 st and at the 28 th day respectively) in contrast to the dose increments observed in our trial [START_REF] Wilder-Smith | Treatment of severe pain from osteoarthritis with slow-release tramadol or dihydrocodeine in combination with NSAID's: a randomized study comparing analgesia, antinociception and gastrointestinal effects[END_REF].

QL was evaluated by EORTC QLQ C 30 an instrument with sound psychometric values [START_REF] Aaronson | The European Organisation for Research and Treatment of Cancer QLQ -C 30: A Quality-of-Life Instrument for Use in International Clinical Trials in Oncology[END_REF] adapted in Poland [START_REF] Majkowicz | Metody pomiaru jakosci zycia stosowane w badaniach wlasnych[END_REF]. In functional scales better global QL was associated with DHC and better emotional functioning with tramadol (treatment time effect and interaction of drug and treatment time). Better cognitive functioning was connected with DHC (interaction of drug and treatment time). Better global QL observed in patients treated with DHC might be a consequence of better analgesia, less fatigue, nausea and vomiting, and sleep disturbance in this group. Better emotional functioning in patients treated with tramadol might be connected with a possible antidepressant tramadol effect observed in some experimental studies [START_REF] Yalcin | Antidepressant-like effect of tramadol in the unpredictable chronic mild stress procedure: possible involvement of the noradrenergic system[END_REF] and less constipation [START_REF] Raffa | Opioid and Nonopioid Components Independently Contribute to the Mechanism of Action of Tramadol, an "Atypical" Opioid Analgesic[END_REF]. A trend of better physical functioning in DHC group was noted (p = 0.063 for treatment time effect; p = 0.09 for interaction of drug and treatment time). There were no differences in role (work) and social functioning.

In symptom scales DHC therapy was associated with less pain, fatigue and sleep disturbance (treatment time effect and interaction of drug and treatment time), less nausea and vomiting, better appetite (interaction of drug and treatment time). Less constipation (interaction of drug and treatment time) and less financial problems (treatment time effect) were connected with tramadol therapy. No differences in dyspnoea and diarrhea were observed. Results of EORTC QLQ C 30 symptom scales are similar to those of modified ESAS, which comprised two additional scales for constipation and vomiting assessment [START_REF] Leppert | Ocena analgezji i objawow niepozadanych tramadolu i dihydrokodeiny o kontrolowanym uwalnianiu u chorych z bolem nowotworowym -na podstawie zmodyfikowanej skali ESAS[END_REF]. This is the case for better analgesia and less intense nausea during DHC administration and less constipation during tramadol treatment. The results might be associated with different mode of action of the drugs: tramadol displays significant non-opioid component [START_REF] Raffa | Opioid and Nonopioid Components Independently Contribute to the Mechanism of Action of Tramadol, an "Atypical" Opioid Analgesic[END_REF] while DHC comparing to tramadol has significantly stronger affinity to mu opioid receptors [START_REF] Schmidt | Affinities of Dihydrocodeine and its Metabolites to Opioid Receptors[END_REF] and thus may cause more intense constipation [START_REF] Wilder-Smith | The visceral and somatic antinociceptive effects of dihydrocodeine and its metabolite, dihydromorphine. A cross-over study with extensive and quinidine-induced poor matabolizers[END_REF]. Nausea which is the most common tramadol adverse effect is connected with serotonin level increase [START_REF] Barann | Effects of tramadol and O-demethyl-tramadol on human 5-HT reuptake carriers and human 5-HT3A receptors: A possible mechanism for tramadol-induced early emesis[END_REF]. All patients did not receive prophylactic antiemetic and laxative, which would influence the results. Patients treated with DHC experienced less sleep disturbance, which might be connected with better analgesia and less fatigue. Opposite to ESAS results no differences in dyspnoea measured by EORTC QLQ C 30 were observed in this study. In finances less problems in tramadol group was not connected with the analgesic type (both drugs were provided free of charge to all patients) but rather with patients' social situation.

Differences observed in QL may suggest prescription practice. Prophylactic antiemetic administration in patients starting tramadol and laxative therapy in patients commencing DHC should be considered. Laxatives administration in patients treated with tramadol may rely on the individual assessment of constipation risk as tramadol role is negligible with this regard [START_REF] Wilder -Smith | The analgesic tramadol has minimal effect on gastrointestinal motor function[END_REF]. Similar individual approach may concern antiemetic use when starting DHC. ECOG [START_REF]Functional assessment scale[END_REF] and Karnofsky [START_REF] Karnofsky | The clinical evaluation of chemotherapeutic agents in cancer[END_REF] PS was low and decreased due to general deterioration during the study period. A significant decrease of activity was the treatment time but not drug effect or drug and treatment time interaction. No differences in PS between patients treated with tramadol and DHC were found.

The results of the study should be interpreted with significant caution as we were not able to demonstrate drug effect in any of the EORTC QLQ C 30 functional or symptom scales, ECOG and Karnofsky PS. The treatment period with each analgesic was 7 days only. Another limitation of the study was non-blinded design and lack of wash-out period, which might have influenced the results after drug switch; however the latter approach might be justified by ethical reasons in cancer pain management. Patients with neuropathic pain component were excluded as they usually need strong opioids and adjuvant analgesics administration. The number of patients recruited was small; thus the results should be replicated in a controlled study with longer follow up.

In spite of these limitations to our knowledge this is the first clinical study evaluating the influence of CR tramadol and CR DHC treatment on QL of patients with cancer pain. To conclude in functional scales DHC treatment was associated with better global QL and cognitive functioning and a trend of better physical functioning; tramadol therapy was connected with better emotional functioning; no differences in role and social functioning were found. In symptom scales DHC therapy was associated with better analgesia, less fatigue, sleep disturbances, nausea and vomiting, better appetite while tramadol treatment was connected with less constipation and financial problems; no differences in dyspnoea and diarrhea were found. All these differences were due to treatment time effect or drug and treatment time interaction with no drug influence. No differences in PS between tramadol and DHC treatment were found; in both groups ECOG and Karnofsky PS deteriorated as a result of the treatment time effect with no drug influence observed.

Conflict of

  explore the impact of tramadol and DHC treatment on QL of patients with nociceptive cancer pain.

  in DHC group were observed. A trend towards better physical functioning in DHC group was noted (p = 0.063 for treatment time; p = 0.09 for interaction of drug and treatment time).

Fig. 1 .

 1 Fig. 1. CONSORT diagram of the study

Page 18 of 19 International Journal of Clinical Practice International Journal of Clinical Practice

  Interest Disclosure Statement: We declare no conflict of interest connected with the article. Financial source: The financial support for the study was provided by departmental sources of Poznan University of Medical Sciences.Tab. 3. Results of two-way ANOVA. Dependent variables: functional scales of EORTC QLQ C 30, independent variables: drug influence, time of QL measurement and interaction of drug and treatment time. Degree of freedom: for a drug 1, for the treatment time and interaction 2. Tab. 4. Results of two-way ANOVA. Dependent variables: symptom scales of EORTC QLQ C 30, independent variables: drug influence, time of QL measurement and interaction of drug and treatment time. Degree of freedom: for a drug 1, for the treatment time and interaction 2. Tab. 5. Performance status (ECOG and Karnofsky) statistical characteristics (mean, standard error) and the comparison of performance status at baseline with the treatment period Tab. 6. Results of two-way ANOVA. Dependent variables: ECOG and Karnofsky preformance status, independent variables: drug influence, treatment time effect and interaction of drug and treatment time. Degree of freedom for a drug 1, for the treatment time and interaction 2

	Page 13 of 19	International Journal of Clinical Practice
	Tab. 1. Equianalgesic doses of CR tramadol and CR DHC when drugs Tab. 2. Pain intensity (VAS) at baseline (before analgesics
	switched after 7 days of the treatment administration, day 0), during tramadol and DHC treatment (days 1 -14;
	mean and standard deviation). Higher scores mean more intense pain
	Tramadol CR		DHC CR
	Day of the	Tramadol	DHC	p-value*
	2 x 100 mg treatment Performance Functional scales of Symptom scales of status scale Performance	2 x 60 mg Drug and Mean, p* influence Treatment time Drug Performance treatment time Drug Treatment Drug and status standard error Treatment time Interaction of
	F o r P F o r P F o r P 2 x 150 mg 2 x 200 mg F o r P 0 1 2 3 63.07 ± 13.48 38.33 ± 19.27 33.40 ± 23.86 38.53 ± 20.93 EORTC QLQ C 30 EORTC QLQ C 30 influence assessment status scale Drug influence F o r P Physical functioning P > 0.240 Role functioning (work) P > 0.2 F o r Fatigue P > 0.6 F o r 0 1 F o r ECOG P > 0.316 F o r ECOG P Nausea and vomiting p = 0.072 P 2 P Karnofsky P > 0.516 P	2 x 90 mg 2 x 120 mg 57.73 ± 8.40 10.20 ± 7.73 11.47 ± 11.61 8.13 ± 9.96 effect interaction time effect treatment time effect drug and P > 0.2 0.001 0.003 0.001 P = 0.063 P = 0.09 P > 0.1 interaction p < 0.036 p < 0.01 2.90 ± 0.18 -3.10 ± 0.18 0.008 treatment time P < 0.001 P > 0.162 P > 0.1 P > 0.486 p < 0.001 3.20 ± 0.18 0.001 P < 0.001 P = 0.087
	e e r R e e r R e e r R 2 x 300 mg CR -controlled release tablets e e r R 4 5 6 35.40 ± 19.83 35.87 ± 18.86 31.20 ± 22.10 e e r R Cognitive functioning P > 0.783 Emotional functioning P > 0.508 e e r R Pain P > 0.558 Dyspnoea P > 0.353 e e r R 0 e e 1 e r e r Karnofsky 2 R R	2 x 180 mg 7.80 ± 10.12 7.80 ± 9.64 8.07 ± 8.92 0.001 0.001 0.001 P > 0.1 p = 0.05 p < 0.002 p < 0.013 p < 0.001 p < 0.001 P > 0.222 P > 0.156 5.03 ± 0.18 -4.90 ± 0.19 0.062 4.77 ± 0.20 0.001
	e v i e w e v i e w e v i e w e v i e w 38.20 ± 16.27 30.93 ± 11.83 8.80 ± 9.41 16.67 ± 13.72 e v i e w P > 0.571 P > 0.260 P> 0.827 p < 0.001 e v i e w p = 0.06 p < 0.001 P > 0.174 P > 0.26 e v i e w e e ECOG: Higher scores mean worse PS 7 Analgesic switch 8 Social functioning Global quality of life Sleep Appetite Karnofsky: Higher scores mean better PS v i e v i e Performance status assessment: w w	0.001 0.005 P > 0.260 p < 0.006 p < 0.005 p < 0.039
	9 10 11 Constipation Diarrhea 0 -at baseline 1 -at the 7 th day of the treatment 30.93 ± 10.16 27,60 ± 8,81 28.67 ± 10.93 P > 0.636 P > 0.118 2 -at the 14 th day of the treatment	O n l y O n l y O n l y O n l y 15.20 ± 15.24 18,73 ± 16,95 18.80 ± 16.23 0.002 0.087 0.062 O n l y O n l y P > 0.222 p < 0.001 P > 0.437 O n l O n l O n l P = 0.09 y y y
	12 Finances	35.47 ± 16.18 P > 0.796	12.87 ± 12.46 p < 0.023	0.001 P = 0.092
	13	32.33 ± 20.45	11.33 ± 12.30	0.002
	14	37.93 ± 22.63	11.93 ± 13.70	0.001
		International Journal of Clinical Practice 7