Ecological

Complexity

Volume

Part

A

December

Pages

Original

Research

Article

Effect

of

global

warming

on

sea

level

rise

A

modeling

study

JB

Shukla

a

Maitri

Verma

b

AK

Misra

b

Get

rights

and

content

Highlights

A

nonlinear

model

is

proposed

to

study

the

response

of

sea

level

to

global

warming

The

effect

of

sea

level

rise

on

the

population

dynamics

is

also

investigated

Stability

analysis

is

carried

out

to

determine

the

longterm

behavior

of

system

Numerical

simulation

explores

the

effect

of

key

parameters

on

the

system

dynamics

Abstract

Global

mean

sea

level

has

been

rising

in

response

to

global

warming

since

the

past

few

decades

and

is

anticipated

to

potentially

affect

the

coastal

population

The

main

driver

of

global

warming

is

the

enhanced

concentration

of

the

heattrapping

gas

carbon

dioxide

in

the

atmosphere

In

this

paper

we

propose

a

nonlinear

mathematical

model

to

study

the

effect

of

an

increase

in

the

anthropogenic

carbon

dioxide

emissions

on

sea

level

rise

and

its

effect

on

the

human

population

The

longterm

behavior

of

the

proposed

system

is

analyzed

using

stability

theory

of

differential

equations

The

model

analysis

shows

that

an

increase

Keywords

Mathematical

model

Atmospheric

carbon

dioxide

Global

warming

Sea

level

rise

Stability

Introduction

Global

warming

is

affecting

all

of

us

in

many

ways

Sea

level

rise

is

one

of

the

direst

impacts

of

global

warming

which

is

affecting

the

coastal

population

of

many

countries

Global

warming

is

a

result

of

the

increase

in

radiative

forcing

of

climate

system

due

to

the

enhanced

anthropogenic

emissions

of

greenhouse

gases

primarily

carbon

dioxide

CO

Human

activities

between

and

have

caused

nearly

increase

in

the

atmospheric

concentration

of

carbon

dioxide

This

upsurge

in

carbon

dioxide

concentration

has

caused

a

radiative

forcing

of

watts

per

square

meter

which

is

nearly

of

the

total

radiative

forcing

of

all

the

wellmixed

greenhouse

gases

Stocker

et

al

The

positive

radiative

forcing

created

by

carbon

dioxide

and

other

greenhouse

gases

has

increased

the

Earths

energy

budget

leading

to

increase

in

the

global

surface

temperature

The

global

mean

surface

temperature

has

risen

by

nearly

C

since

preindustrial

times

and

is

expected

to

rise

in

future

with

the

increase

in

atmospheric

concentration

of

CO

and

other

greenhouse

gases

A

doubling

of

carbon

dioxide

concentration

is

estimated

to

bring

nearly

C

rise

in

global

mean

surface

temperature

in

comparison

to

preindustrial

level

Lindsey

The

rise

in

global

mean

temperature

leads

to

melting

of

glaciers

and

ice

caps

resulting

in

sea

level

rise

It

is

found

that

the

global

mean

sea

level

has

risen

approximately

m

between

and

with

a

rate

of

nearly

millimeter

per

year

Church

et

al

The

rate

of

sea

level

rise

is

expected

to

increase

in

future

with

the

further

increase

in

atmospheric

temperature

The

fifth

assessment

report

of

IPCC

Intergovernmental

Panel

on

Climate

Change

projects

that

the

global

mean

sea

level

may

rise

by

m

by

the

end

of

century

The

NOAA

National

Oceanic

and

Atmospheric

Administration

projection

of

sea

level

rise

is

up

to

m

by

The

ice

sheets

across

the

world

hold

sufficient

water

to

raise

the

sea

level

by

several

meters

Antarctica

and

Greenland

ice

sheets

are

the

two

largest

ice

sheets

containing

nearly

of

the

freshwater

ice

on

the

Earth

It

is

estimated

that

if

the

Greenland

ice

sheet

melts

completely

sea

level

may

rise

about

m

while

the

melting

of

Antarctic

ice

sheet

may

cause

a

sea

level

rise

of

about

m

It

is

estimated

that

Antarctica

ice

sheet

alone

can

contribute

more

than

a

meter

of

sea

level

rise

by

the

year

and

more

than

m

by

the

year

DeConto

and

Pollard

Sea

level

rise

has

potentially

harmful

consequences

such

as

coastal

flooding

erosion

saltwater

contamination

of

fresh

water

supplies

increase

in

salinity

of

agricultural

soil

hurricane

storm

surge

etc

Nearly

million

people

residing

in

lowelevation

coastal

zones

are

at

risk

of

exposure

to

these

hazards

of

sea

level

rise

Neumann

et

al

Hazards

of

sea

level

rise

may

force

a

portion

of

the

coastal

population

to

migrate

from

coastal

zones

to

inland

zones

affecting

the

coastal

population

dynamics

Sea

level

rise

affects

not

only

the

dynamics

of

regional

population

but

also

the

global

population

by

affecting

the

net

carrying

capacity

of

human

population

due

to

the

reduction

in

inhabitable

land

area

The

elevated

sea

level

has

already

begun

to

wipe

out

many

coastal

areas

For

instance

nearly

half

of

Bhola

Island

in

Bangladesh

has

become

permanently

flooded

due

to

sea

level

rise

leaving

climate

refugees

Penna

and

Rivers

As

most

of

the

global

population

is

residing

in

coastal

zones

and

many

of

the

worlds

major

cities

such

as

New

York

Los

Angeles

California

Sydney

etc

are

located

in

lowlying

coastal

zones

the

future

sea

level

rise

is

likely

to

potentially

influence

the

population

dynamics

From

the

above

it

is

pointed

out

that

the

global

warming

caused

by

an

increase

in

anthropogenic

carbon

dioxide

emissions

may

affect

the

sea

water

level

which

in

turn

may

affect

the

human

population

levels

To

address

the

problem

of

sea

level

rise

it

is

crucial

to

comprehend

the

role

of

anthropogenic

carbon

dioxide

emissions

and

the

associated

temperature

rise

behind

the

sea

level

rise

For

this

purpose

mathematical

modeling

is

an

effective

tool

The

effect

of

climate

change

on

sea

level

rise

is

usually

evaluated

by

using

the

processbased

numerical

models

Gregory

et

al

Meehl

et

al

and

semiempirical

models

Gornitz

et

al

Rahmstorf

Vermeer

and

Rahmstorf

Grinsted

et

al

These

models

are

either

numerical

or

statistical

models

and

use

quantitative

methods

to

assess

the

response

of

sea

level

to

the

temperature

change

and

rely

completely

on

data

Apart

from

these

numerical

and

semiempirical

models

dynamical

models

that

utilize

qualitative

methods

may

also

be

used

to

gain

an

indepth

understanding

of

the

effect

of

temperature

change

on

sea

level

rise

In

recent

years

various

differential

equation

models

have

been

proposed

to

address

the

dynamics

of

greenhouse

gases

and

pollutants

in

the

atmosphere

Tennakone

Naresh

and

Sundar

Naresh

et

al

Caetano

et

al

Sundar

Misra

and

Verma

Misra

and

Verma

Misra

et

al

Shukla

et

al

Verma

and

Misra

In

these

studies

qualitative

analysis

of

the

proposed

models

is

carried

out

to

assess

the

longterm

effect

of

various

crucial

factors

on

the

dynamics

of

greenhouse

gases

and

pollutants

In

the

present

study

we

formulate

a

nonlinear

mathematical

model

comprising

a

set

of

differential

equations

to

study

the

effect

of

the

enhanced

concentration

of

carbon

dioxide

and

the

associated

temperature

rise

on

the

sea

level

rise

and

the

effect

of

sea

level

rise

on

the

human

population

The

qualitative

analysis

of

the

proposed

model

is

performed

to

assess

the

longterm

behavior

of

the

system

using

stability

theory

of

differential

equation

Model

formulation

Human

activities

increase

the

concentration

of

carbon

dioxide

gas

in

the

atmosphere

An

increase

in

the

concentration

of

carbon

dioxide

gas

causes

an

increase

in

the

global

mean

temperature

and

it

causes

melting

of

glaciers

and

ice

caps

leading

to

sea

level

rise

As

the

sea

level

rises

the

land

area

submerged

in

water

increases

Due

to

the

decrease

in

the

inhabitable

land

area

carrying

capacity

of

human

population

declines

To

model

this

scenario

we

consider

six

dynamical

variables

namely

the

human

population

N

t

the

atmospheric

concentration

of

carbon

dioxide

C

t

the

average

surface

temperature

T

t

the

mass

of

ice

sheets

I

c

t

the

mean

sea

level

W

t

and

the

land

area

submerged

in

sea

L

t

It

is

assumed

that

the

carbon

dioxide

enters

in

the

atmosphere

due

to

natural

processes

as

well

as

due

to

human

activities

The

natural

source

of

carbon

dioxide

includes

volcanoes

decomposition

fire

ocean

release

respiration

etc

The

anthropogenic

CO

emission

consists

mainly

of

the

postindustrial

CO

emissions

from

fossil

fuel

burning

industrial

processes

deforestation

and

other

human

related

activities

The

carbon

dioxide

is

uptaken

by

the

natural

sinks

such

as

forests

oceans

fossil

fuels

carbonated

rocks

etc

Prior

to

the

industrial

revolution

the

atmospheric

concentration

of

CO

had

been

relatively

stable

as

the

natural

source

and

sinks

of

carbon

dioxide

were

relatively

balanced

and

the

anthropogenic

emissions

were

insignificant

relative

to

natural

variability

IPCC

Let

C

be

the

equilibrium

concentration

of

carbon

dioxide

in

the

absence

of

anthropogenic

carbon

dioxide

emissions

or

preindustrial

equilibrium

concentration

of

carbon

dioxide

Since

the

industrial

revolution

anthropogenic

emission

of

carbon

dioxide

has

increased

linearly

with

the

increase

in

human

population

Newell

and

Marcus

Onozaki

so

the

emission

rate

of

carbon

dioxide

from

anthropogenic

sources

is

taken

to

be

proportional

to

human

population

λN

The

emission

of

CO

from

the

natural

sources

is

assumed

to

be

a

constant

Q

The

uptake

rate

of

carbon

dioxide

by

the

natural

sinks

is

assumed

to

be

proportional

to

the

atmospheric

concentration

of

CO

ie

λ

C

Nikolskii

Let

T

be

the

average

surface

temperature

in

the

absence

of

anthropogenic

carbon

dioxide

emissions

or

the

preindustrial

average

surface

temperature

An

increase

in

the

concentration

of

carbon

dioxide

leads

to

an

increase

in

the

atmospheric

temperature

It

is

observed

that

the

increase

in

global

average

temperature

has

linear

correlation

with

the

increase

in

CO

concentration

IPCC

Hence

it

is

assumed

that

the

temperature

increases

at

a

rate

proportional

to

the

increase

in

the

concentration

of

CO

due

to

human

activities

We

have

assumed

that

the

ice

sheets

gain

mass

at

a

constant

rate

q

The

increase

in

temperature

causes

melting

of

ice

sheets

leading

to

sea

level

rise

In

the

study

by

Rahmstorf

the

rate

of

sea

level

rise

is

taken

to

be

proportional

to

the

increase

in

temperature

However

the

rate

of

sea

level

rise

also

depends

on

the

melting

of

ice

sheets

Meehl

et

al

Hence

we

assume

that

the

ice

sheets

melt

at

a

rate

βI

c

T

T

where

β

is

the

melting

rate

coefficient

of

ice

sheets

The

melting

of

ice

sheets

causes

an

increase

in

the

water

level

at

a

rate

γβI

c

T

T

where

γ

is

a

proportionality

constant

which

represents

the

conversion

of

ice

sheets

into

sea

level

rise

The

surging

sea

level

causes

coastal

erosion

and

inundation

leading

to

an

increase

in

the

submerged

area

In

some

studies

it

is

found

that

the

longterm

shoreline

retreat

rate

is

proportional

to

sea

level

rise

Leatherman

et

al

Zhang

et

al

In

view

of

this

it

is

assumed

that

the

submerged

area

increases

at

a

rate

proportional

to

the

increase

in

sea

water

level

ie

ϕ

W

W

where

ϕ

is

the

growth

rate

coefficient

of

the

submerged

area

and

W

is

the

sea

water

level

when

the

temperature

is

T

As

the

submerged

area

increases

the

land

inhabitable

to

human

population

decreases

and

this

negatively

affects

the

growth

of

human

population

via

decreasing

the

carrying

capacity

of

the

human

population

at

a

rate

r

N

L

L

where

r

is

the

declination

rate

coefficient

of

human

population

due

to

increase

in

the

submerged

area

Under

the

above

considerations

the

dynamics

of

the

problem

is

governed

by

the

following

system

of

differential

equations

q

β

T

γ

β

T

W

ϕ

W

L

rN

L

λ

N

C

θ

C

T

where

C

Q

λ

I

c

W

W

L

L

N

C

C

and

T

T

The

description

of

parameters

of

model

system

and

their

unit

are

given

in

Table

In

Table

Mm

stands

for

square

megameter

and

ppm

stands

for

parts

per

million

Description

of

model

parameters

and

their

unit

Parameter

Description

Units

q

Rate

of

formation

of

ice

sheets

teraton

per

year

α

Natural

depletion

rate

coefficient

of

ice

sheets

per

year

β

Melting

rate

coefficient

of

ice

sheets

per

C

per

year

T

Average

surface

temperature

in

the

absence

of

anthropogenic

CO

emissions

C

γ

Conversion

factor

of

sheet

melt

into

sea

level

rise

meter

per

teraton

γ

Natural

depletion

rate

coefficient

of

sea

level

per

year

W

Mean

sea

level

when

the

average

surface

temperature

is

T

meter

ϕ

Growth

rate

coefficient

of

the

submerged

area

due

to

sea

level

rise

Mm

per

meter

per

year

ϕ

Declination

rate

coefficient

of

the

submerged

area

per

year

L

Submerged

land

area

when

sea

level

is

W

Mm

r

Intrinsic

growth

rate

of

human

population

per

year

K

Carrying

capacity

of

human

population

million

persons

r

Declination

rate

of

carrying

capacity

of

human

population

due

to

submerging

of

land

area

in

sea

per

Mm

per

million

persons

per

year

Q

Natural

emission

rate

of

carbon

dioxide

ppm

per

year

λ

Anthropogenic

emission

rate

coefficient

of

CO

ppm

per

million

persons

per

year

λ

Depletion

rate

coefficient

of

carbon

dioxide

per

year

C

Equilibrium

concentration

of

carbon

dioxide

in

the

absence

of

anthropogenic

CO

emissions

ppm

θ

Growth

rate

coefficient

of

the

average

surface

temperature

C

per

ppm

per

year

θ

Depletion

rate

coefficient

of

the

average

surface

temperature

per

year

The

dynamics

of

the

coastal

population

is

assumed

to

be

governed

by

the

following

equation

where

N

c

N

In

the

above

equation

the

constants

s

and

s

are

the

growth

rate

and

natural

death

rate

coefficients

of

the

coastal

population

The

constant

s

is

the

migration

rate

coefficient

of

the

coastal

population

due

to

sea

level

rise

Further

it

is

assumed

that

the

coastal

population

increases

at

a

rate

proportional

to

the

total

population

Once

the

dynamics

of

the

system

is

determined

dynamics

of

the

coastal

population

can

be

obtained

using

Eq

We

assume

that

s

s

so

that

the

coastal

population

can

not

exceed

the

total

population

Lemma

The

set

where

and

is

positively

invariant

and

for

all

satisfying

the

initial

conditions

of

the

model

system

I

c

t

W

t

L

t

N

t

C

t

T

t

Γ

as

t

The

set

Γ

is

called

region

of

attraction

for

the

model

system

For

the

proof

of

the

above

lemma

see

Appendix

A

Model

analysis

A

qualitative

analysis

of

the

nonlinear

system

is

carried

out

using

the

stability

theory

of

differential

equations

The

stability

theory

addresses

the

stability

properties

of

the

solutions

of

a

dynamical

system

In

the

following

we

examine

the

stability

of

the

stationary

solutions

ie

the

equilibrium

points

of

the

model

system

Equilibria

The

model

system

exhibits

the

following

two

nonnegative

equilibrium

points

i

ii

E

I

c

W

L

N

C

T

The

existence

of

equilibrium

E

is

obvious

The

components

of

interior

equilibrium

E

can

be

obtained

by

solving

the

following

set

of

algebraic

equations

From

we

have

From

and

we

have

From

and

we

have

From

and

we

have

Using

Eq

in

we

have

Using

Eq

in

we

get

the

following

quadratic

equation

in

N

From

we

find

that

F

and

F

K

Thus

there

exists

a

unique

positive

root

of

Eq

say

N

in

the

interval

K

which

is

given

as

Using

the

above

value

of

N

in

Eqs

and

we

get

the

positive

values

of

C

T

W

and

L

respectively

The

equilibrium

level

of

coastal

population

is

given

by

Remark

It

is

found

that

and

Thus

the

model

predicts

that

the

equilibrium

levels

of

surface

temperature

sea

water

level

and

land

area

submerged

in

water

increase

while

the

equilibrium

levels

of

ice

sheet

mass

coastal

population

and

total

human

population

decrease

as

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

increases

Remark

It

is

found

that

and

This

shows

that

the

equilibrium

levels

of

surface

temperature

and

sea

water

level

decrease

and

the

equilibrium

levels

of

ice

sheet

mass

coastal

population

and

total

human

population

increase

as

the

depletion

rate

coefficient

of

carbon

dioxide

increases

Remark

It

is

found

that

and

showing

that

the

equilibrium

level

of

land

area

submerged

in

water

increases

whereas

the

equilibrium

levels

of

total

human

population

and

coastal

population

decline

with

an

increase

in

the

melting

rate

coefficient

of

ice

sheets

Stability

of

equilibria

The

local

stability

of

the

equilibrium

E

is

determined

by

observing

the

sign

of

eigenvalues

of

the

Jacobian

matrix

J

of

the

model

system

evaluated

at

the

equilibrium

E

The

eigenvalues

of

J

are

found

to

be

α

γ

ϕ

r

λ

and

θ

Since

one

of

the

eigenvalues

of

J

is

r

which

is

positive

the

equilibrium

E

has

locally

unstable

manifold

in

N

direction

Thus

the

system

will

never

approach

to

the

equilibrium

E

We

determine

the

local

as

well

as

global

stability

behavior

of

the

equilibrium

E

by

using

the

Lyapunovs

stability

theory

The

results

obtained

from

stability

analysis

of

E

are

stated

in

the

form

of

following

theorems

Theorem

The

equilibrium

E

is

locally

asymptotically

stable

provided

that

the

following

inequality

is

satisfied

For

the

proof

of

the

above

theorem

see

Appendix

B

Theorem

The

equilibrium

E

is

globally

asymptotically

stable

in

Γ

provided

that

the

following

inequality

is

satisfied

For

the

proof

of

the

above

theorem

see

Appendix

C

Numerical

simulation

To

check

the

feasibility

of

analysis

and

to

depict

the

effect

of

key

parameters

on

the

dynamics

of

the

system

we

have

conducted

numerical

simulation

using

MATLAB

The

snow

accumulation

rates

for

the

Antarctica

and

Greenland

ice

sheets

are

gigatons

per

year

and

gigatons

per

year

respectively

Steffen

et

al

Thus

the

rate

of

formation

of

ice

sheets

q

is

taken

to

be

teratons

or

million

gigatons

per

year

We

have

taken

r

per

year

and

λ

ppm

per

million

persons

per

year

from

Verma

and

Misra

The

carrying

capacity

of

human

population

K

is

taken

to

be

million

which

is

according

to

the

United

Nations

Population

Divisions

mediumvariant

scenario

UNPD

The

atmospheric

lifetime

of

carbon

dioxide

is

years

Jacobson

thus

the

value

of

λ

lies

in

the

range

per

year

We

take

C

ppm

the

preindustrial

CO

concentration

IPCC

To

decide

the

value

of

other

parameters

of

the

model

system

we

use

the

annual

time

series

data

of

world

population

carbon

dioxide

concentration

global

average

temperature

and

mean

sea

level

The

annual

data

of

atmospheric

carbon

dioxide

for

the

period

is

taken

from

NOAAESRL

data

set

NOAA

while

that

of

world

population

for

the

period

is

obtained

from

UNPD

We

have

taken

the

annual

data

of

the

global

average

surface

temperature

for

the

period

of

from

NOAA

The

annual

time

series

data

of

the

mean

sea

level

observed

at

a

particular

station

Reykjavik

Iceland

between

the

years

is

taken

from

NOAA

We

assume

the

value

of

W

m

which

is

m

below

the

level

The

preindustrial

temperature

is

taken

to

be

T

C

We

calibrate

the

model

system

for

different

values

of

α

β

γ

γ

ϕ

ϕ

L

θ

θ

r

and

λ

and

find

the

best

fit

for

α

per

year

β

per

C

per

year

γ

m

per

teraton

γ

per

year

ϕ

Mm

per

meter

per

year

ϕ

per

year

L

Mm

θ

C

per

ppm

per

year

θ

per

year

r

per

Mm

per

million

persons

per

year

and

λ

per

year

The

actual

data

and

the

model

projections

are

illustrated

in

Fig

In

the

absence

of

annual

time

series

data

of

the

land

area

submerged

in

water

and

mass

of

ice

sheets

the

above

parameter

values

provide

a

very

rough

estimate

However

we

try

to

be

as

realistic

as

possible

while

selecting

the

parameter

values

The

parameter

values

used

in

simulation

for

the

model

system

and

are

listed

as

q

β

γ

ϕ

r

K

λ

θ

s

For

the

above

set

of

parameter

values

the

interior

equilibrium

of

system

is

obtained

as

E

The

eigenvalues

of

the

Jacobian

matrix

for

the

model

system

calculated

at

the

equilibrium

E

are

i

i

and

All

the

eigenvalues

are

either

negative

or

having

negative

real

part

showing

that

the

equilibrium

E

is

locally

asymptotically

stable

The

global

stability

condition

is

also

satisfied

for

the

set

of

parameter

values

showing

that

the

equilibrium

E

is

globally

asymptotically

stable

in

Γ

Download

highres

image

Download

fullsize

image

Fig

Comparison

between

model

fit

and

actual

data

of

atmospheric

concentration

of

carbon

dioxide

human

population

average

surface

temperature

and

mean

sea

level

In

Fig

we

have

plotted

the

solution

trajectories

in

I

c

W

L

and

N

C

T

spaces

for

different

initial

starts

inside

the

region

of

attraction

Γ

From

this

figure

it

can

be

observed

that

all

the

solution

trajectories

starting

inside

the

region

of

attraction

finally

approach

to

and

N

C

T

respectively

This

shows

the

global

stability

of

E

in

the

region

of

attraction

Γ

in

I

c

W

L

and

N

C

T

spaces

Fig

is

plotted

to

show

the

effect

of

an

increase

in

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

ie

λ

on

the

equilibrium

levels

of

average

surface

temperature

mean

sea

level

land

area

submerged

in

water

and

human

population

From

this

figure

it

can

be

observed

that

an

increase

in

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

leads

to

increase

in

the

equilibrium

levels

of

temperature

sea

water

and

the

land

area

submerged

in

water

while

the

equilibrium

level

of

the

human

population

decreases

due

to

increase

in

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

This

happens

because

as

the

anthropogenic

emission

rate

of

carbon

dioxide

increases

the

concentration

of

atmospheric

carbon

dioxide

increases

which

leads

to

increase

Fig

From

this

figure

we

can

observe

that

as

the

growth

rate

of

human

population

increases

the

equilibrium

levels

of

surface

temperature

and

sea

water

increase

Download

highres

image

Download

fullsize

image

Fig

The

global

stability

of

E

inside

the

region

of

attraction

Γ

in

I

c

W

L

and

N

C

T

spaces

Download

highres

image

Download

fullsize

image

Fig

Variation

of

surface

temperature

sea

level

land

area

submerged

in

sea

water

and

human

population

with

respect

to

time

for

different

values

of

the

anthropogenic

emission

rate

of

carbon

dioxide

λ

Download

highres

image

Download

fullsize

image

Fig

Variation

of

surface

temperature

and

sea

level

with

respect

to

time

for

different

values

of

the

intrinsic

growth

of

human

population

r

Fig

shows

that

as

the

melting

rate

coefficient

of

icesheets

ie

β

increases

the

equilibrium

level

of

land

area

submerged

in

water

increases

while

that

of

human

population

decreases

Fig

is

plotted

to

show

the

effect

of

changes

in

parameters

λ

β

ϕ

and

s

on

the

equilibrium

level

of

coastal

population

This

figure

shows

that

the

coastal

population

settles

to

a

lower

level

as

the

values

of

parameters

λ

β

ϕ

and

s

increase

The

sea

level

rise

may

be

controlled

by

cutting

down

the

carbon

dioxide

emissions

as

well

as

by

increasing

the

depletion

rate

of

carbon

dioxide

To

show

the

effects

of

changes

in

the

anthropogenic

emission

rate

and

depletion

rate

of

carbon

dioxide

on

the

equilibrium

levels

of

atmospheric

temperature

and

sea

water

we

have

plotted

Fig

This

figure

shows

that

the

atmospheric

temperature

and

sea

water

level

approach

low

values

for

low

emission

rates

and

high

depletion

rates

of

carbon

dioxide

while

for

high

emission

rates

and

low

depletion

rates

of

carbon

dioxide

the

temperature

and

sea

water

levels

approach

higher

values

Download

highres

image

Download

fullsize

image

Fig

Variation

of

surface

temperature

and

sea

level

with

respect

to

time

for

different

values

of

the

melting

rate

coefficient

of

ice

sheets

β

Download

highres

image

Download

fullsize

image

Fig

Variation

of

coastal

population

with

respect

to

time

for

different

values

of

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

λ

melting

rate

coefficient

of

icesheets

β

growth

rate

coefficient

of

submerged

area

due

to

sea

level

rise

ϕ

and

migration

rate

coefficient

of

the

coastal

population

s

Download

highres

image

Download

fullsize

image

Fig

Effect

of

change

in

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

λ

and

depletion

rate

coefficient

of

carbon

dioxide

λ

on

the

equilibrium

levels

of

surface

temperature

and

sea

water

level

To

assess

that

how

sensitive

the

model

solutions

are

to

change

in

some

of

the

key

parameters

λ

β

λ

and

ϕ

a

basic

differential

sensitivity

analysis

of

the

equations

in

the

model

system

is

performed

with

respect

to

these

parameters

following

Bortz

and

Nelson

The

sensitivity

function

of

a

state

variable

X

with

respect

to

a

parameter

w

is

given

by

We

have

calculated

the

semirelative

sensitivity

solutions

ie

those

represent

the

change

in

state

of

the

system

on

doubling

of

a

parameter

value

These

solutions

are

plotted

in

Fig

From

this

figure

we

can

see

that

the

doubling

of

parameter

β

has

significant

influence

over

the

mass

of

ice

sheets

I

c

t

sea

level

W

t

land

area

submerged

in

water

L

t

and

human

population

N

t

while

the

change

in

parameters

λ

and

λ

significantly

influence

all

the

variables

The

change

in

parameter

ϕ

has

large

influence

over

L

t

and

N

t

while

its

influence

over

other

variables

is

negligible

From

the

second

plot

of

Fig

we

can

see

that

doubling

of

parameter

β

causes

an

increase

of

m

in

the

sea

level

while

the

doubling

of

the

parameter

λ

leads

to

a

rise

of

m

in

the

sea

level

over

a

period

of

years

The

doubling

of

parameter

λ

causes

a

decrease

of

m

in

the

sea

level

over

a

period

of

years

From

the

fourth

plot

of

Fig

we

can

see

that

the

doubling

of

parameter

ϕ

causes

a

decrease

of

million

in

the

human

population

over

a

period

of

years

Download

highres

image

Download

fullsize

image

Fig

Semirelative

sensitivity

solutions

for

the

state

variables

with

respect

to

the

parameters

λ

the

anthropogenic

emission

rate

coefficient

of

carbon

dioxide

β

the

melting

rate

coefficient

of

icesheets

λ

the

depletion

rate

coefficient

of

carbon

dioxide

and

ϕ

the

growth

rate

coefficient

of

submerged

land

area

due

to

sea

level

rise

Conclusion

Acknowledgements

Authors

are

thankful

to

the

handling

editor

and

the

anonymous

reviewers

for

their

useful

suggestions

which

improved

the

quality

of

the

paper

The

second

author

is

thankful

to

National

Board

of

Higher

Mathematics

Department

of

Atomic

Energy

Government

of

India

for

providing

financial

support

in

form

of

the

postdoctoral

fellowship

No

Appendix

A

Proof

of

Lemma

We

prove

the

lemma

following

Freedman

and

So

First

of

all

we

show

that

the

set

Γ

is

positively

invariant

ie

for

all

I

c

W

L

N

C

T

Γ

I

c

t

W

t

L

t

N

t

C

t

T

Γ

for

all

t

From

the

fourth

equation

of

the

model

system

we

have

Comparing

the

above

differential

inequality

with

differential

equation

and

using

a

standard

comparison

theorem

Hale

we

get

N

t

K

for

all

t

From

the

fifth

equation

of

the

model

system

we

get

This

implies

that

C

t

t

Now

from

the

sixth

equation

of

the

model

system

we

get

This

implies

that

T

t

t

Proceeding

same

as

above

from

the

first

three

equations

of

the

model

system

it

can

be

found

that

Now

we

will

show

that

for

all

satisfying

the

initial

conditions

of

the

model

system

I

c

t

W

t

L

t

N

t

C

t

T

t

Γ

as

t

From

the

fourth

equation

of

the

model

system

we

have

Noting

the

solution

of

and

using

a

standard

comparison

theorem

Hale

we

have

Let

ϵ

be

given

Then

there

exists

a

t

such

that

Hence

From

the

fifth

equation

of

the

model

system

we

get

As

before

there

exist

t

t

such

that

Hence

Similarly

from

the

last

equation

of

the

model

system

we

find

that

there

exist

t

t

such

that

Hence

Proceeding

same

as

above

from

the

first

three

equations

of

the

model

system

we

get

t

W

t

and

L

t

Appendix

B

Proof

of

Theorem

In

the

following

we

obtain

the

characterization

of

the

local

stability

of

the

interior

equilibrium

E

using

the

Lyapunovs

direct

method

La

Salle

and

Lefschetz

We

first

linearize

the

model

system

about

E

by

using

the

following

transformations

L

L

l

N

N

n

C

C

c

and

T

T

p

where

i

c

w

l

N

c

and

p

are

small

perturbations

about

E

Now

we

consider

the

following

positive

definite

function

where

m

m

m

m

and

m

are

positive

constants

to

be

chosen

suitably

Differentiating

V

with

respect

to

t

along

the

solutions

of

the

linearized

system

of

we

get

β

β

p

γ

β

w

γ

β

p

w

ϕ

l

w

n

l

λ

n

c

θ

pc

dV

dt

will

be

negative

definite

provided

the

following

conditions

hold

We

choose

For

this

selection

of

m

m

m

and

m

the

inequalities

and

are

satisfied

Now

from

the

rest

of

inequalities

we

can

select

m

provided

the

condition

holds

Appendix

C

Proof

of

Theorem

In

the

following

we

derive

sufficient

conditions

for

the

global

stability

of

the

interior

equilibrium

E

in

Γ

by

using

the

Lyapunovs

direct

method

Consider

the

following

positive

definite

function

U

where

k

k

k

k

and

k

are

positive

constants

to

be

chosen

suitably

Differentiating

U

with

respect

to

t

along

the

solutions

of

the

model

system

we

get

β

β

T

γ

β

W

γ

β

T

W

ϕ

L

W

N

N

L

λ

N

C

θ

T

C

dU

dt

will

be

negative

definite

inside

Γ

provided

the

following

inequalities

are

satisfied

We

choose

For

this

selection

of

k

k

k

and

k

the

inequalities

and

are

satisfied

Now

from

the

rest

of

inequalities

we

can

select

k

provided

the

condition

holds

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Sensitivity

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