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A Structured Prey-Predator System with Alternative
Diffusive Predator Resource
Rachana Pathak1, Anuj Kumar2and Manju Agarwal 3
Department of Mathematics and Astronomy,
Lucknow University, Lucknow-226007, India
1rachanapathak2@gmail.com
2guptaanujkm89@gmail.com
3manjuak@yahoo.com
ABSTRACT
The present paper deals with a stage-structured prey-predator model having alternative
diffusive resource for predator.The mathematical analysis with regard positivity, bounded-
ness, equilibria and their stability with and without diffusion, and bifurcation are carried out.
Using differential inequality, we obtain sufficient conditions that ensure the persistence of
the system.Finally, analytical results are verified with the help of numerical simulations.
Keywords: Alternative Resource;Prey-Predator;Diffusion; Stability; Persistence.
2000 Mathematics Subject Classification: 49K05, 49K15, 49S05.
1 Introduction
The dynamics of prey-predator systems is one of the dominant subjects in mathematical ecol-
ogy due to its universal existence and importance.These interactions show a slight difference in
stability when a resource (such as predator’s resource) is present in the system, as it works as
an alternative source of intake besides the prey population. Such system has been considered
broadly, by several mathematicians, ecologists and biologists(Dubey and Das,2000,Dubey and
Upadhyay,2004;Freedman and Shukla,1989;Freedman and Krisztin,1992;He,1996;Sarkar and
Roy,1993;Shi,2009;Wang and Chen,1997) .There are many species in the nature, whose in-
dividual members have a life history that takes them through two stages : immature and
mature.Agarwal and Devi(2010) studied the effects of toxicants on the dynamics of a single
species growth model with stage structure consisting of immature and mature stages and also
showed that equilibrium level of immature population is more prone for the effects of toxicants
in comparison to the mature population.Agarwal and Devi (2010) proposed and analysed a
ratio-dependent predator-prey model where the prey population is stage structured and preda-
tor population is influenced by the resource biomass.They noted that the influence of resource
biomass on the predator population may lead to the extinction of prey population at a lesser
value of maturity time in comparison to the absence of the resource biomass.
International Journal of Ecological Economics and Statistics;
Volume. 38, Issue No. 3; Year 2017; Int. J. Ecol. Econ. Stat.;
ISSN 0973-1385 (Print), ISSN 0973-7537 (Online)
Copyright © 2017, [International Journal of Ecological Economics & Statistics]
ISSN 0973-1385 (Print), ISSN 0973-7537 (Online)
www.ceser.in/ceserp
www.ceserp.com/cp-jour
Keeping above all in mind we have developed a resource based prey-predator model. The dif-
fusion of the predator’s resource has also been taken into account. In the model, the stability
of the system has been analysed by using the theory of nonlinear ordinary differential equa-
tions. The prey population is taken to be structured according to two stages immature and
mature. The model has initially been analysed locally and globally without diffusion. We have
also studied the diffusivity coefficient with initial boundary conditions. Finally, it is concluded
that the large value of the diffusion coefficient makes the uniform steady state stable in both
the cases when the interior equilibrium is stable or when it is unstable without diffusion.
2 Mathematical Model
Let us consider a prey-predator system with alternative diffusive predator resource. In the
model, the prey is structured according to two different stages, immature xi(t, u, v)and mature
xm(t, u, v)population respectively.y(t, u, v)is the density of the predator. R(t, u, v )denotes the
resource present at time t≥0and coordinates (u, v)∈S, where Sis a simply connected
domain in u−vplane.Let the specific rate of resource uptake per unit predator in unit time
is hR and supply rate of the external resource input to the system be constant and equal to
R0. Resource involved in predator growth, may be used due to consumption at a rate hRy
and its density also reduces due to certain degradation factors present in the environment
at a rate aR. Predator lose certain part of its density, at a rate gdue to death. The uptake
concentration of mature prey by predator is taken as of linear form at the rate l. Assuming that
at any time t>0birth into the immature prey population is proportional to the existing mature
prey population with proportionality constant αand the death rate of immature prey population
is proportional to that existing immature population, with proportionality constant γ. For mature
prey population, the death rate is taken to be logistic in nature, with proportionality constant
β. Finally, observing that those immature preys born at time t−τthat survive to time texit
from the immature category and enter the mature prey population.Keeping in view of these
considerations, the non-linear model is proposed as follows
∂xi
∂t =αxm−γxi−αe−γτxm(t−τ),
∂xm
∂t =αe−γτ xm(t−τ)−βx2
m−lxmy
1+R+xm,
∂y
∂t =lxmy
1+R+xm+hRy −gy2,
∂R
∂t =(R0−aR)R−hRy +dR2R,
,(2.1)
Imposing the following boundary and initial conditions on the system 2.1 as,
xi(0,u,v)=ψi(u, v )≥0,
xm(0,u,v)=ψm(u, v )≥0,
y(0,u,v)=θ(u, v )≥0,
R(0,u,v)=ψ(u, v )≥0,
(2.2)
∂xi
∂n =∂xm
∂n =∂y
∂n =∂R
∂n =0,(2.3)
xi(0) ≥0,x
m(t)=φm(t)≥0,−τ≤t<0,y(0) ≥0,R(0) ≥0.(2.4)
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88
where nis the unit outward normal to the region ∂S .ψi,ψ
m,θand ψare smooth initial functions.
Sis simply connected domain in the u−vplane with piecewise smooth boundary ∂S .∇2≡
∂2
∂u2+∂2
∂v2is the Laplacian diffusion operator.dRis the co efficient of R.
Here we first analyse model 2.1.For continuity of initial conditions,we require
xi(0) = 0
−τ
αeγsφm(s)ds, (2.5)
the total survivors of these prey members who were born between −τand 0.With the help of
2.5,the solution of the first equation of system 2.1 can be written in terms of solutions of xmas
follows
xi(t)=t
t−τ
αe−γ(t−s)xm(s)ds, (2.6)
Now from equations 2.5 and 2.6, we can noted that mathematically no information of xi(t)is
needed for the system 2.1.Using the equations 2.5 and 2.6,we can find the properties of the xi
if we know the properties of xm.Now we consider the following model for analysis
∂xm
∂t =αe−γτ xm(t−τ)−βx2
m−lxmy
1+R+xm,
∂y
∂t =lxmy
1+R+xm+hRy −gy2,
∂R
∂t =(R0−aR)R−hRy +dR2R,
,(2.7)
boundary and initial conditions on the system 2.7 as,
xm(0,u,v)=ψm(u, v )≥0,y(0,u,v)=θ(u, v)≥0,R(0,u,v)=ψ(u, v)≥0,(2.8)
∂xm
∂n =∂y
∂n =∂R
∂n =0,x
m(t)=φm(t)≥0,−τ≤t<0,y(0) ≥0,R(0) ≥0.(2.9)
3 Analysis of the model system without Diffusion
Model 2.1 without diffusion takes the following form
dxm
dt =αe−γτ xm(t−τ)−βx2
m−lxmy
1+R+xm
,
dy
dt =lxmy
1+R+xm
+hRy −gy2,
dR
dt =(R0−aR)R−hRy,
(3.1)
3.1 Positivity of Solutions
Theorem 3.1. All solutions of the system 3.1 are positive for all t≥0.
P roof Clearly y(t)and R(t)for y(0) >0,R(0) >0,t > 0.Now xm(0) >0, hence if there
exists t0such that xm(t0)=0, then t0>0. Assume that t0is the first time such that xm(t)=0,
that is t0=inf {t>0:xm(t)=0}. Then
˙xm(t0)=αe−γτ φm(t0−τ)>0,(0 ≤t0≤τ)
αe−γτ φm(t0−τ)>0,(t0>τ),(3.2)
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89
so that ˙xm(t0)>0.Hence for sufficiently small ε>0,˙xm(t0−ε)>0.But by the definition of
t0,˙xm(t0−ε)≤0,a contradiction. Hence xm(t)>0for all t≥0.
3.2 Boundedness of Solutions
In the following lemma, we state the bounds of the various variables which would be needed
in our study.
Lemma 3.2. The set Ω={(xm,y,R):0≤xm≤A, 0≤y≤A2and0≤R≤A1},is the re-
gion of attraction for all solutions initiating in the interior of the positive octant, where A=
αe−γτ
β,A
2=lA +hA1
gand A1=R0
a.
P roof Let (xm,y,R)be solution with positive initial values (xm0,y
0,R
0). From system 3.1, we
get
dxm
dt ≤αe−γτ xm(t−τ)−βx2
m.(3.3)
According to comparison principle, it follows that
lim
t→∞ sup xm≤αe−γτ
β=A. (3.4)
Now from the system 3.1, we get
dR
dt ≤R0R−aR2.(3.5)
According to comparison principle again, we get
lim
t→∞ sup R≤R0
a=A1.(3.6)
From the second equation of the system 3.1, we get
dy
dt =(lA +hA1)y−gy2.(3.7)
Again using comparison principal, we have
lim
t→∞ sup y≤lA +hA1
g=A2.(3.8)
This completes the proof of lemma.
3.3 Equilibrium Analysis
The analysis of the system 3.1, we can get the following equilibria:
1. E0(0,0,0), this equilibrium point is obvious.
2. E1(xm1,0,0), where xm1=αe−γτ
β.
3. E2(0,0,R
2), where R2=R0
a.
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90
4. E3(xm3,0,R
3), where xm3=αe−γτ
βand R3=R0
a, obtained by the equation 2.7.
5. E4(0,y
4,R
4), where y4=R0h
ag +h2and R4=R0g
ag +h2, obtained by the equation 2.7.
6. E5(¯xm,¯y, 0), existence of equilibrium point E5is given as.
Here ¯xmand ¯yare the positive solutions of the system of algebraic equations given below
αe−γτ −β¯xm−l¯y
1+ ¯xm=0,
l¯xm
1+ ¯xm−g¯y=0.(3.9)
From second equation of the system 3.9 we get
¯y=l¯xm
g(1 + ¯xm)(3.10)
Putting the value of ¯yfrom equation 3.10 in the first equation of the system 3.9 we have
following expression
F(¯xm)=gβ ¯
x3
m+2βg −gαe−γτ¯
x2
m+gβ −2gαe−γτ +l2¯xm−gαe−γτ .(3.11)
From equation (3.11), we have
F(0) = −gαe−γτ <0.(3.12)
And
Fαe−γτ
β=l2αe−γτ
β>0.(3.13)
Thus there exists a ¯xm,0<¯xm<αe−γτ
βsuch that F(¯xm).Now, the sufficient condition
for the uniqueness of E5is F(¯xm)>0.From 3.11 we can find and F(¯xm)as follows,
F(¯xm)=3gβ ¯
x2
m+22βg −gαe−γτ¯xm+gβ −2gαe−γτ +l2>0.(3.14)
By using 3.11 in 3.14 we can check that F(¯xm)is positive. Hence the existence of E5.
7. E6(x∗
m,y
∗,R
∗)existence of this equilibrium point given as.
Here x∗
m,y
∗and R∗are the positive solutions of the system of algebraic equations
αe−γτ −βx∗
m−ly∗
1+R∗+x∗
m=0,
lx∗
m
1+R∗+x∗
m+hR∗−gy∗=0,
R0−aR∗−hy∗=0.
(3.15)
From last equation of the system 3.15,we have
y∗=R0−aR∗
h.(3.16)
with condition
R0>aR
∗.(3.17)
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91
From first equation of the system 3.15, we get following expression
R∗=p1x∗2
m+p2x∗
m+p3
p4+p5x∗
m
.(3.18)
where p1=βh,p2=h(β−αe−γτ ),p
3=R0l−hαe−γτ ,
p4=al +hαe−γτ ,p
5=−βh.
Using the value of y∗and R∗in the second equation of the system 3.15, we have
G(x∗
m)=b1x∗4
m+b2x∗3
m+b3x∗2
m+b4xm+b5.(3.19)
where
b1=p1h2+ag(p1+p5),
b2=p2
5(hl −gR0)+h2+ag(q1+p1p4+p2p5)+h2−gR0+agp1p5,
b3=−gR0p2
5+q4+hlq4+h2+ag(q2+p2p4+p3p5)+h2−gR0+ag(p1p4+p2p5),
b4=−gR0p2
4+q4+hlp2
4+h2+ag(q3+p3p4)+h2−gR0+ag(p2p4+p3p5),
b5=(p3+p4)h2+agp3−gR0p4,
q1=2p1p2,q
2=p2
2+2p1p3,q
3=2p2p3,q
4=2p4p5.
From equation 3.19, we have
G(0) = (p3+p4)h2+agp3−gR0p4<0.(3.20)
if h2+agp3<gR
0p4.
And
Gαe−γτ
β=b1βαe−γτ
β4
+b2αe−γτ
β3
+b3αe−γτ
β2
+b4αe−γτ
β+b5>0.
(3.21)
Thus there exists a x∗
m,0<x
∗
m<αe−γτ
βsuch that G(x∗
m).Now, the sufficient condition
for the uniqueness of E6is G(x∗
m)>0.From 3.20 we can find and G(x∗
m)as follows,
G(x∗
m)=4b1x3
m+3b2x2
m+2b3xm+b4>0.(3.22)
By using 3.20 in 3.22 we can check that G(x∗
m)is positive.
Hence the existence of E6.
3.4 Local Stability Analysis
To discuss the local stability of the system 3.1,we compute the variational matrices correspond-
ing to each equilibrium point.The entries of general variational matrix are given by differentiat-
ing the right side of system 3.1 with respect to xm,y and R.From these matrices we note the
following results:
1. The equilibrium point E0is unstable manifold in xm−Rplane.
International Journal of Ecological Economics & Statistics
92
2. The equilibrium points E1and E2are unstable points in y−Rand xm−yplane respec-
tively.
3. The equilibrium point E3and E4is unstable in y−and xm−direction respectively.
4. The equilibrium point E5is unstable in xm−direction if αe−(γ+λ)τ(a+R0)g+h2>
hlR0in presence of τand in absence of τ,α(a+R0)g+h2>hlR
0.
5. The characteristic polynomial corresponding to interior equilibrium point is given as:
λ3+l1λ2+l2λ+l3+e−λτ l4λ2+l5λ+l6=0.(3.23)
where
l1=−a22 −a33 +g11;l2=−a12 a21 −a23a32 +a22 a33 −a22g11 −a33 g11,
l3=−a13a21a32 +a12a21 a33 −a23a32 g11 +a22a33 g11;l4=−αe−γτ ;l5=−αe−γτ (a22 +a33 ),
l6=αe−γτ (a23a32 −a22a33),
a12 =−lxm
1+R+xm,a
13 =lxmy
(1+R+xm)2,a
21 =(1+R)ly
(1+R+xm)2,a
22 =lxm
1+R+xm+hR −2gy,
a23 =hy −lxmy
(1+R+xm)2,a
32 =−hR,
a33 =R0−2aR −hy, g11 =2βxm−(1+R)ly
(1+R+xm)2.
CaseA :τ=0The associated characteristic equation of the system 3.1 is
λ3+(l1+l4)λ2+(l2+l5)λ+l3+l6=0.(3.24)
By using Routh-Hurwitz criteria all roots of equation 3.24 have negative real parts if and only if
.
(H1):l1+l4>0,l
2+l5>0and Δ=(l1+l4)(l2+l5)−(l3+l6)>0.So the equilibrium point
E6is locally asymptotically stable when H1holds.
CaseB :τ== 0 Let λ=iω be one such that.Substituting this in equation 3.23 and equating
real and imaginary parts, we get
l5ωcosωτ −l6−l4ω2sinωτ =ω3−l2ω. (3.25)
l6−l4ω2cosωτ +l5ωsinωτ =l1ω2−l3.(3.26)
Squaring and adding the equations 3.25 and 3.26, we get
ω6+l2
1−l2
4−2l2ω4+l2
2−l2
5−2l1l3+2l4l6ω2+l2
3−l2
6=0.(3.27)
We define
H(ω)=ω6+l2
1−l2
4−2l2ω4+l2
2−l2
5−2l1l3+2l4l6ω2+l2
3−l2
6and assume that
International Journal of Ecological Economics & Statistics
93
H2:H(0) = l3<l
6holds.
It is easy to check that H(0) <0if H2holds and H(∞)>0.Therefore, the equation 3.27
always has at least one positive root.
Again solving equations 3.25 and 3.26, we get a critical value of delay given as follows
τ=1
ωcos−1l6−l4ω2l1ω2−l3−l5ω2l2−ω2
l2
5ω2+(l6−l4ω2)2+2kπ
δ,k =0,1,2.... (3.28)
3.5 Hopf Bifurcation
Differentiating equation 3.23 with respect to τ,we obtained
dλ
dτ −1
=−3λ2+2l1λ+l2+e−λτ (2l4+l5)
λ(λ3+l1λ2+l2λ+l3)−τ
λ.(3.29)
We can obtain here d(Reλ)
dτ τ=τ0
=0.(3.30)
Verifying numerically it has been obtained that the transversality condition holds and hence
Hopf bifurcation occurs at τ=τ0.
3.6 Global Stability Analysis
Theorem 3.3. Let the following inequality holds in the region Ω
max βx∗
m(1 + R∗+x∗
m)
3(1+x∗
my∗+R∗),g, a(1 + R∗+x∗
m)
x∗
m(1 + y∗)>l
2,(3.31)
and
τ>1
γlog α
M1,
where
M1=βx∗
m−3l(1 + x∗
my∗+R∗)
2(1+R∗+x∗
m),
then the equilibrium point E6is globally asymptotically stable.
proof For proving this theorem, we will show that
lim
t→∞
xm(t)=x∗
m,lim
t→∞
y(t)=y∗,lim
t→∞
R(t)=R∗.
Let us consider following positive definite function
V=1
2(xm−x∗
m)2+y−y∗−y∗ln y
y∗+R−R∗−R∗ln R
R∗.(3.32)
The derivative of Vwith respect to time tis given as
dV
dt =αe−γτ (xm−x∗
m)(xm−x∗
m)(t−τ)−β(xm+x∗
m)−lx∗
my∗
(1 + R+xm)(1+ R∗+x∗
m)(xm−x∗
m)2
−g(y−y∗)2−a(R−R∗)2+−lx∗
m
(1 + R+xm)(1+ R∗+x∗
m)(y−y∗)(R−R∗)+ l(1 + R∗)
(1 + R+xm)(1+ R∗+x∗
m)
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(xm−x∗
m)(y−y∗)+ lx∗
my∗
(1 + R+xm)(1+R∗+x∗
m)(xm−x∗
m)(y−y∗).(3.33)
Applying Cauchy-Schwartz inequality,we get following expression
dV
dt ≤1
2αe−γτ (xm−x∗
m)2+1
2αe−γτ (xm−x∗
m)2(t−τ)−g(y−y∗)2−a(R−R∗)2
−β(xm+x∗
m)−lx∗
my∗
(1 + R+xm)(1+R∗+x∗
m)(xm−x∗
m)2+1
2
lx∗
m
(1 + R+xm)(1+R∗+x∗
m)(R−R∗)2
+1
2
lx∗
m
(1 + R+xm)(1+R∗+x∗
m)(y−y∗)2+1
2
l(1 + R∗)
(1 + R+xm)(1+R∗+x∗
m)(xm−x∗
m)2
+1
2
l(1 + R∗)
(1 + R+xm)(1+R∗+x∗
m)(y−y∗)2+1
2
lx∗
my∗
(1 + R+xm)(1+R∗+x∗
m)(xm−x∗
m)2
+1
2
lx∗
my∗
(1 + R+xm)(1+R∗+x∗
m)(R−R∗)2.(3.34)
Arranging similar terms,maximizing right hand side and assuming following inequality
max βx∗
m(1 + R∗+x∗
m)
3(1+x∗
my∗+R∗),g, a(1 + R∗+x∗
m)
x∗
m(1 + y∗)>l
2.We get
dV
dt ≤−M1(xm−x∗
m)2−M2(y−y∗)2−M3(R−R∗)2+1
2αe−γτ (xm−x∗
m)2+1
2αe−γτ (xm−x∗
m)2(t−τ).
(3.35)
where
M1=βx∗
m−3l(1 + x∗
my∗+R∗)
2(1+R∗+x∗
m),M
2=g−l
2,M
3=a−lx∗
m(1 + y∗)
2(1+R∗+x∗
m).Now we assume
following Lyapunov function
V1=V+αe−γτ
2t
t−τ
(xm−x∗
m)2(θ)dθ, (3.36)
and,hence
dV1
dt =dV
dt +αe−γτ
2(xm−x∗
m)2(t)−αe−γτ
2(xm−x∗
m)2(t−τ).(3.37)
Substituting the value of dV
dt in equation 3.37,we have
dV1
dt ≤−M1(xm−x∗
m)2−M2(y−y∗)2−M3(R−R∗)2+αe−γτ (xm−x∗
m)2.(3.38)
Therefore
dV1
dt ≤−M1−αe−γτ (xm−x∗
m)2−M2(y−y∗)2−M3(R−R∗)2.(3.39)
The last expression is negative definite provided that
τ>1
γlog α
M1.
Using the application of the Lyapunov-LaSalle type theorem (Ma etal.,2009) we get required
result
lim
t→∞
xm(t)=x∗
m,lim
t→∞
y(t)=y∗,lim
t→∞
R(t)=R∗.
This completes the proof of the theorem(3.3).
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95
3.7 Permanence of Solutions
Theorem 3.4. Assume that αe−γτ >lA
2and R0>hA
2. Then system 3.1 is permanent.
P roof
From first equation of the system 3.1,we get
dxm
dt ≥αe−γτ xm(t−τ)−lA2xm−βx2
m,, (3.40)
According to comparison principle,it follows that
xmmin =αe−γτ −lA2
β=A3.(3.41)
with condition αe−γτ >lA
2.Similarly way, we get from the second equation of the system 3.1.
dy
dt ≥lA3
1+A1+A+hA1y−gy2.. (3.42)
By using comparison principle,we get
ymin =1
glA3
1+A1+A+hA1=A4.(3.43)
From the last equation the system 3.1 we get
dR
dt ≥(R0−hA2)R−aR2.(3.44)
Again using comparison principle,we have
Rmin =R0−hA2
a.(3.45)
with condition R0>hA
2.Hence the Theorem follows.
4 Model with Diffusion
In this section considering the model 2.7 and noting that as a sequence xm=x∗
m,y =y∗,R =
R∗are the uniform steady points for this system.Here, it is required to show that if E6is asymp-
totically stable for system 3.1, then the corresponding uniform steady state is asymptotically
stable for system 2.7.Further, the cases where E6is unstable for the system 3.1,but the corre-
sponding uniform steady state is uniform for 2.7 is discussed.This is supported by choosing a
positive definite function.
W(t)=S
V(xm,y,R)dA. (4.1)
Where Vis same as defined in 3.32.Differentiating Wwith respect to time along the solutions
of model 2.7,we obtain
dW
dt =I1+I2.(4.2)
Where
I1=S
dV
dt dA. (4.3)
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96
I2=Sd1
∂V
∂xm
∇2xm+d2
∂V
∂y ∇2y+dR
∂V
∂R∇2RdA. (4.4)
Since dV
dt is negative definite,I1is also the same sign as dV
dt .
The Vconsists of the following properties:
∂V
∂xm
|∂S =∂V
∂y |∂S =∂V
∂R|∂S =0 (4.5)
for all points of S.
∂2V
∂R∂xm
=∂2V
∂y∂xm
=0.(4.6)
∂2V
∂R2>0,∂2V
∂x2
m
>0,∂2V
∂y2>0.(4.7)
as d1=d2=0.So,the term which is left in I2is
I2=S
dR
∂V
∂R 2R. (4.8)
Using Green’s First Identity in the plane
S
F1∇2G1dA =S
F1
∂G1
∂n dS1−S
(∇F.∇G)dA. (4.9)
Now equation 4.8 can be written as
I2=−S
dR
∂2V
∂R2∂R
∂u 2
+∂R
∂v 2dA. (4.10)
We concluded from the above analysis as follows:
1. if dV
dt is negative definite then dW
dt is also negative definite which gives that if the equi-
librium point E6is asymptotically global stable in absence of diffusion, then the corre-
sponding uniform steady state E6of boundary value problem 2.7 must also be globally
stable.
2. If dV
dt is positive definite then also dW
dt can be made negative definite with the increasing
value of dRwhich implies that if E6is unstable in the absence of diffusion then the cor-
responding uniform steady stable of 2.7 can be made stable by the increment made in
diffusion coefficient (i.e.dR).
Now,dealing with the case that the region under consideration Sis rectangle given by S=
{(u, v):0≤u≤δ1,0≤v≤δ2}
From 4.10, I2can be rewritten as,
I2=−dRS
∂2V
∂R2∂(R−R∗)
∂u 2
+∂(R−R∗)
∂v 2dA (4.11)
where dA =dudv
δ2
0δ1
0∂(R−R∗)
∂u 2
dudv =1
δ1δ2
01
0∂(R−R∗)
∂v 2
du1dv. (4.12)
International Journal of Ecological Economics & Statistics
97
where u=u1δ1.Utilizing the following inequality (Denn,1975)in 4.11 we get
1
0∂x
∂u2
du ≥π21
0
x2du. (4.13)
δ2
0δ1
0∂(R−R∗)
∂u 2
dudv ≥π2
δ1δ2
0δ1
0
(R−R∗)2dudv =π2
δ2
1S
(R−R∗)2dA. (4.14)
Similarly
δ2
0δ1
0∂(R−R∗)
∂u 2
dudv ≥π2
δ2
2S
(R−R∗)2dA. (4.15)
Hence dW
dt =I1+I2can be written as
dW
dt ≤S
VdA−δ2
1+δ2
2π2
δ2
1δ2
2dRa2R∗
R2
0S
(R−R∗)2dA.(4.16)
From 4.16, note that if dV
dt is positive definite then also dW
dt could be made negative definite by
increasing the diffusion coefficient dRto a sufficiently large value that is dR>δ2
1δ2
2R2
0
R∗a2π2δ2
1+δ2
2SVdA,
then the uniform steady state is asymptotically stable.
5 Numerical simulation
This section is mainly based on numerical simulation due to the complexity of the analytical
solutions.Using numerical simulation instead of real world data,which of course would be of
great interest,has the advantage to isolate the effects of the species interaction easily.
We use the parameter values as α=8,γ =0.1,τ =0.9,β =0.5,l =2,h =1,g =0.4,R
0=
10,a=0.4.
Here, we also note that all conditions of local stability, global stability and persistence are
satisfied for the above set of parameter values.
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98
Figure 1. Variation of population with time at τ=0.1.
Figure 2.Phase Portrait at τ=0.1.
Figure 3. Variation of population with time at τ=10.
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99
Figure 4.Phase Portrait at τ=10.
Figure(1)and Figure(2) show that system is stable when value of τ=0.1and trajectories of
all population trend to their equilibrium points. When we increase the value of τthe density
of population decrease and trend to zero when value of τbecome 10 (see Figure(3)).Also
dynamic behaviour of the system become unstable when τ=10.The presentation of this
behaviour is shown by Figure (4).
6 Discussion
In this paper, a resource based prey-predator system has been proposed and analysed into
consideration where the prey is being structured, according to the two stages immature and
mature. In modelling process, we assume that the predator has alternative diffusive resource
besides the prey. The diffusion of resource has taken into account, which play an important
role in the dynamics of the system. The analysis of the system shows that diffusion coefficient
can effectively change the stability of the system.
Acknowledgements
Dr.Rachana Pathak thankfully acknowledges the NBHM PDF, Ref. No.(2/40(29)/2014/R&D-
11).
Prof.Manju Agarwal thankfully acknowledges the DST PROJECT, Ref.No.(SR/MS-23: 793/12).
Anuj Kumar thankfully acknowledges the Science and Engineering Research Board (SERB),
New Delhi, India for the financial assistance in the form of Junior Research Fellowship (SR/S4/MS:793/12).
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100
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