Dynamics of a di usive Leslie-Gower predator-prey model with … · 2019. 8. 6. · Dynamics of a...

Annual Review of Chaos Theory, Bifurcations and Dynamical SystemsVol. 8, (2018) 1-20, www.arctbds.com.This journal is published under the Creative Commons Attribution 4.0 International:

Dynamics of a diffusive Leslie-Gower predator-preymodel with nonlinear prey harvesting

R. SivasamyDepartment of Mathematics, SRMV College of Arts and Science (Bharathiar University)

Coimbatore-20, Indiae-mail: [email protected]

K. SathiyanathanDepartment of Mathematics, SRMV College of Arts and Science (Bharathiar University)

Coimbatore-20, Indiae-mail: [email protected]

Abstract: This paper considers a modified Leslie-Gower predator-prey model withCrowley-Martin functional response and nonlinear prey harvesting strategy subject tothe no-flux boundary conditions. To understand the dynamics of the considered system,we derive sufficient conditions for permanence analysis, local stability, global stabilityand Hopf bifurcation of interior equilibrium point. Further we also investigate the exis-tence and non-existence of non-constant positive steady state solutions. Finally numericalsimulations are performed to verify the proposed theoretical results.

Keywords: Predator-Prey Model, Crowley-Martin functional response, Prey har-vesting, Stability and Hopf bifurcation.

Manuscript accepted 08 01, 2018.

1 Introduction

Inspired by the pioneering work of Lotka and Volterra, the dynamics of interactionsbetween two species (predator-prey) models have been receiving much more attentionin the field of both ecology and mathematical ecology [5, 11, 16]. At some practicalsituations, the predators have to go for alternative food when there is low density for its

2 R. Sivasamy, K. Sathiyanathan

favorite food. To cover this issue, Leslie and Gower in [12] have introduced a new modelof the following form:

dn

dt= rn

(1− n

k

)− pg(n, p),

dp

dt= p

(c− dp

n+ l

),

(1)

where n(t) and p(t) are the respective population densities of prey and predator at timet > 0, r and k stand for the per capita growth rate of prey and carrying capacity,respectively. The term g(r, z) is called functional response term which denotes the in takerate of prey by per capita predator and this term plays a major role in deriving the exactmathematical model, for more details one can see [4, 6, 8, 9, 13, 17]. The parameters cand d indicate the growth rate and maximum reduction rate of predator respectively andl represents the measure of extent to which environment provides protection to predator.The dynamics of the model given in (1) have been studied by many researchers in theliterature (see [3, 7, 15, 18] and references cited therein).

The interference among the predators will be common in nature when predators com-pete for food. Hence Crowley-Martin functional response has been introduced by Crowleyand Martin and it is given by g(n, p) = c1n

1+an+bp+abnp, where c1, a, b, are positive parame-

ters that are used for effects of capture rate, handling time and magnitude of interferenceamong predators respectively on the feeding rate. This functional response assumes thatpredator feeding rate will decrease due to high predator density (interference among thepredator individuals) even when prey density is high (presence of handling or searchingof prey by predator individual). Holling II (g(n, p) = c1n

1+an) and Beddington-DeAngelis

(g(n, p) = c1n1+an+bp+abnp

) functional responses are special cases of Crowley-Martin func-tional response. Therefore it is meaningful to introduce Crowley-Martin functional re-sponse in modified Leslie-Gower model which takes the following form

dn

dt= rn

(1− n

k− cp

1 + an+ bp+ abnp

),

dp

dt= p

(c− dp

n+ l

),

(2)

where the parameters have same meaning as discussed above. Ali and Jazar in [2] con-sidered the system (2) in which they derived the sufficient conditions for global stability,Hopf bifurcation and the existence and nonexistence of periodic solutions.

The diffusion terms in predator-prey models are quite common due to the movementof species from higher to lower concentration areas as a result of good living environment,food, etc. The effects of reaction-diffusion term in system (2) have been dealt in [21,24, 19]. They obtained the conditions for the local and global asymptotic stability andpattern formation around the interior equilibrium points. Besides harvesting of speciesis essential in view of economic profit in fishery, forestry, and wildlife management. Theharvested prey-predator model has been constructed and studied in [14, 23, 20] undervarious harvesting strategies such as constant-yield harvesting, constant-effort harvesting,age-selective harvesting etc. Specifically nonlinear harvesting is more realistic from theeconomical as well as biological point of view rather than other strategies [20]. The

Dynamics of a diffusive Leslie-Gower predator-prey model 3

nonlinear harvesting takes the following form

H(z) =qEz

m1E +m2z,

where z is the density of prey (or predator), q stands for the catchability coefficient, Erepresents the external effort applied for harvesting and m1, m2 are positive constants.To the authors’ knowledge, it is evident that prey harvesting has not yet been consideredin system (2) with diffusion term. This fact has motivated our present study.

Hence we consider the modified Leslie-Gower predator-prey model with Crowley-Martin functional response and nonlinear prey harvesting in the form:

∂N

∂T= RN

(1− N

K

)− mNP

1 + AN +BP + ABNP−H(N) +D1∇2N,

∂P

∂T= P

(C − DP

N + L

)+D2∇2P,

(3)

for x ∈ Ω, t > 0. Now we make the following non-dimensional scheme N → Kn, P →Rmp, T → 1

Rt, and let α = AK, β = BR

m, d1 = D1

R, g = qE

m2KR, h = m1E

m2K, γ = C

R, δ =

DRmKC

, ρ = LK, d2 = D2

R. Then the system (3) becomes

∂n

∂t= n

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

)+ d1∇2n, x ∈ Ω, t > 0,

∂p

∂t= γp

(1− δp

n+ ρ

)+ d2∇2p, x ∈ Ω, t > 0,

n(x, 0) = n0 ≥ 0, p(x, 0) = p0 ≥ 0, x ∈ Ω,∂∗n = ∂∗p = 0, x ∈ ∂Ω, t > 0,

(4)

where the positive constants d1 and d2 are diffusion coefficients of prey and predatorrespectively, ∇2 = ∂2

∂x2+ ∂2

∂y2is usual two dimensional Laplacian operator in variable

X ≡ (x, y) ∈ Ω ⊂ R2 and Ω is a bounded domain in R2 with smooth boundary ∂Ω. ∂∗indicates the outward unit normal vector of the boundary ∂Ω. Thus the prey and predatordensities at location X and time t are denoted by n(X, t) and p(X, t) respectively. Theinitial data n(x, 0) ≥ 0 and p(x, 0) ≥ 0 are continuous functions. The zero flux boundaryconditions assure that there is no fluxes of populations through the boundary, that is, noexternal input is imposed from outside.

2 Permanence

In this section, we present the permanence analysis of the system (4). For this, weintroduce the basic definition and lemmas in the following:

Definition 1 System (4) is said to be permanent if there exist positive constants c and csuch that (n(x, t), p(x, t)) of (4) with n0 ≥ 0 and p0 ≥ 0 satisfy

0 < c ≤ limt→∞

inf minx∈Ω

n(x, t) ≤ limt→∞

sup maxx∈Ω

n(x, t) ≤ c,

0 < c ≤ limt→∞

inf minx∈Ω

p(x, t) ≤ limt→∞

sup maxx∈Ω

p(x, t) ≤ c.


Lemma 2 Suppose that z(x, t) satisfies

∂z

∂t− d∆z = z(1− z), x ∈ Ω, t > 0,

z(x, 0) = z0(x) ≥ 0, x ∈ Ω,

∂∗n = 0, x ∈ ∂Ω, t > 0.

Then limt→∞ z(x, t) = 1 for any x ∈ Ω.

Theorem 3 Let (n(x, t), p(x, t)) be the solution of (4) with n0 ≥ 0 and p0 ≥ 0. It holds

1. n(x, t) ≥ 0, p(x, t) ≥ 0, ∀t > 0, x ∈ Ω, and

2. limt→∞

sup maxx∈Ω

n(x, t) ≤ 1 and limt→∞

sup maxx∈Ω

p(x, t) ≤ 1 + ρ

δ.

Proof. It follows from the first equation of (4) that n satisfies

∂n

∂t− d1∆n ≤ n(1− n), x ∈ Ω, t > 0.

Then, by Lemma 2, for any ε > 0, there exists t1 > 0 such that

n(x, t) ≤ 1 + ε, x ∈ Ω, t ≥ t1 (5)

which implies

limt→∞

sup maxx∈Ω

n(x, t) ≤ 1.

Similarly, from second equation of (4), we have

∂p

∂t− d2∆p = γp

(1− δp

n+ ρ

), x ∈ Ω, t > 0.

Hence there exists a constant t2 > t1 such that

p(x, t) ≤ (1 + ε+ ρ)

δ+ ε, t ≥ t2.

Therefore, by arbitrariness of ε, we complete the proof.

Theorem 4 Let (n(x, t), p(x, t)) be the solution of (4) with n0 ≥ 0 and p0 ≥ 0. If h > gand bh > h+ gb, we have

limt→∞

inf minx∈Ω

n(x, t) ≥ (h− g)δ + (βh− h− gβ)(1 + ρ)

hδ + βh(1 + ρ)

limt→∞

inf minx∈Ω

p(x, t) ≥ (h− g)δ + hδρ+ (βh− h− gβ + βhρ)(1 + ρ)

hδ + βh(1 + ρ).

(6)


Proof. From the first equation of (4) and Theorem 3, we have

∂n

∂t− d1∆n ≥ n

(1− n− p

1 + βp− g

h

), t > t2,

where p = 1+ε+ρδ

+ ε. Since h > g and βh > h + gβ hold, for any ε > 0, there exists aconstant t3 > t2 such that

n ≥ n =(h− g)δ + (βh− h− gβ)(1 + ρ)

hδ + βh(1 + ρ)− ε > 0, t ≥ t3 (7)

holds. It follows from the second equation of (4) and (7) that

∂p

∂t− d2∆p ≥ γp

(1− δp

n+ ρ

), t > t3.

So there exists a constant t4 ≥ t3 such that

p ≥ p =n+ ρ

δ− ε > 0, t > t4, (8)

for ε > 0 small enough. From (7) and (8), we obtain (6).Let c = minn, p and c = max1, δ−1(1+ρ). Then, by Definition 1, Theorems 3 and

4, we arrive at the following result.

Theorem 5 If h > g and βh > h+ gβ hold, then system (4) is permanent.

3 Equilibria and stability analysis

3.1 Constant equilibria

The equilibria of system (4) are given by

n

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

)= 0,

γp

(1− δp

n+ ρ

)= 0.

Solving the above equations, we get the following equilibrium points:

i. the trivial equilibrium point E0 = (0, 0).

ii. The predator free axial equilibrium point

E1 =

(1− h

2+

1

2

√(1− h)2 − 4(g − h), 0

).

iii. The prey extinction equilibrium point E1 = (0, ρδ).


iv The interior equilibrium point E∗ = (n∗, p∗), where p∗ = n∗+ρδ

with n∗ is a root ofthe following cubic equation in z

az4 + bz3 + cz2 + dz + e = 0, (9)

where

a = −αβ,b = (1− h)αβ − (β + αδ + αβρ),

c = −(δ + βρ) + (1− h)(β + αδ + αβρ) + (h− g)αβ − 1,

d = (h− g)(δ + βρ)− (h− g)(β + αδ + αβρ)− (ρ+ h),

e = (h− g)(δ + βρ)− ρh.

Remark 6 The equilibrium E0 always exists and when g < h, the equilibrium E1 exists.It is easy to observe from (9) that the leading coefficient a is always negative and e ispositive if

(h− g)(δ + βρ) > ρh (10)

holds. Hence, if (10) is satisfied, the Descartes rule of sign assures that the equation (9)possesses at least one positive root. Further equation (9) has a unique positive root, sayn∗, if (10) holds along with any one the following conditions:

H1 b < 0, c < 0 and d < 0

H2 b < 0, c < 0 and d > 0

H3 b < 0, c > 0 and d > 0

H4 b > 0, c > 0 and d > 0.

Hereafter we always assume that the system (4) satisfies any one of the above conditions.

3.2 Local stability analysis

In this subsection, we study the local stability of the system (4) about the positive equi-librium point E∗. For the notation simplicity, we set:

z = (n, p)T , D = diagd1, d2,

G(z) =

[g1

g2

]=

n

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

)γp

(1− δp

n+ ρ

) .

Then the system (4) can be rewritten in the vector form as

∂z

∂t= D∆z +G(z). (11)


The linearization of (11) at any point E(n, p) is given by

∂z

∂t−D∆z = Gz(E)z, (12)

where

Gz(E) =

(g11 g12

g21 g22

)=

γ − n

(1 + αn)(1 + βp)2

γδp2

(n+ ρ)2γ − 2γδp

(n+ ρ)

,

with γ = 1−2u− p

(1 + αn)2(1 + βp)− gh

(h+ n)2. The Jacobian matrix of (12) at E∗ takes

the form

Gz(E∗) =

(g∗11 g∗12

g∗21 g∗22

)=

γ∗ − n∗

(1 + αn∗)(1 + βp∗)2

γ

δ−γ

,

where γ∗ = n∗(

αp∗

(1 + αn∗)2(1 + βp∗)+

g

(h+ n∗)2− 1

). Let µi, ϕi be an eigenpair of

the operator −∆ on Ω with Neumann boundary condition. 0 = µ1 < µ2 < · · ·E(µi) is theeigenspace corresponding to µi in C1(Ω) and ϕij, j = 1, 2, · · · dimE(µi) is an orthonormalbasis of E(µi). Also let

W =

(n, p)T ∈ [C2(Ω) ∩ C1(Ω)]2|

∣∣∣∣∂n∂n =∂p

∂n= 0

(13)

and Wij = cϕij|c ∈ R2. Consider the following decomposition

W =∞⊕i=1

Wi, (14)

where Wi =⊕dimE(µi)

j=1 Wij with Wij as the eigenspace corresponding to µi.

Remark 7 The stability of equilibrium points E0 and E1 is calculated by eigenvalues ofJacobian matrix at corresponding equilibrium points. By simplest calculations, E0 and E1

are always unstable.

Theorem 8 Assume that

1 >αp∗

(1 + αn∗)2(1 + βp∗)+

g

(h+ n∗)2. (15)

Then positive equilibrium point E∗ is locally asymptotically stable.

Proof. For each i ≥ 1, Wi is invariant under the operator L = D∆ + Gz(E∗) and λ is

an eigenvalue of L on Wi if and only if it is an eigenvalue of the matrix −µiD +Gz(E∗).

DenoteA(µi, E

∗) = −µiD +Gz(E∗). (16)


The characteristic equation of A(µi, E∗) is

λ2 − tr[A(µi, E∗)]λ+ detA(µi, E

∗) = 0, (17)

where

trA(µi, E∗) = −µi(d1 + d2) + g∗11 + g∗22

= −µi(d1 + d2) + γ∗ − γ,detA(µi, E

∗) = d1d2µ2i − (d1g

∗22 + d2g

∗11)µi + detGz(E

∗)

= d1d2µ2i − (−d1γ + d2γ

∗)µi − γγ∗ −γn∗

δ(1 + αn∗)(1 + βp∗)2.

From condition (15), it is easy to check that detA(µiE∗) > 0 and trA(µiE

∗) < 0, fori ≥ 1. So the two characteristic eigenvalues λ1i, λ2i of A(µiE

∗) have negative real partsfor i ≥ 1. Hence the proof is complete.

3.3 Hopf bifurcation

Here we derive the conditions for Hopf bifurcation near E∗.

Theorem 9 Assume that γ∗ = γ,

n∗

δ(1 + αn∗)(1 + βp∗)2> γ∗ (18)

andd1 > d2. (19)

Then the system (4) exhibits Hopf bifurcation near E∗.

Proof. If γ∗ = γ, it is evident that trA(µ1, E∗) = 0, tr[A(µi, E

∗)] < 0, as well asdetA(µi, E

∗) > 0 for i ≥ 2. Hence, when γ∗ = γ, A(µ1, E∗) has a pair of purely imaginary

eigenvalues for i = 1 and negative real parts for i ≥ 2. Moreover transitivity condition isgiven by d trA(µ1,E∗)

dγ= −1 6= 0. Then, if the conditions (18)and (19) hold, then the system

(4) has a family of periodic solutions bifurcating from E∗ when γ is around γ∗. Hence theproof.

3.4 Global stability analysis

In this subsection, we derive the condition for global stability of the equilibrium point E∗.In this regard, we construct the following Lyapunov function

V (n, p) =

∫Ω

n− n∗ − n∗ log

( nn∗

)+

1

γδ

(p− p∗ − p∗ log

(p

p∗

))dΩ, (20)

where (n(x, t), p(x, t)) is any solution of the system (4). For simplicity, we denote

P =αp∗

ρ(1 + αn∗)(1 + βp∗)− 1

(1 + α)(1 + ρ)(δ + β(1 + ρ))(1 + βp∗)

+1

4(1 + βp∗)2.


Theorem 10 Assume that

αp∗

(1 + αn∗)(1 + βp∗)+

g

h(h+ n∗)≤ 1 +

1

1 + ρ(21)

and1

1 + ρ− 1

4δ2ρ2≥ P +

g

ρh(h+ n∗). (22)

Then the positive equilibrium point E∗ is globally asymptotically stable.

Proof. Now taking the derivative of V with respect to t along the trajectory of thesystem (4), we have

dV

dt= I1 + I2

=

∫Ω

d1

(n− n∗

n

)∆n+

d2

γδ

(p− p∗

p

)∆p

dΩ

+

∫Ω

(n− n∗)

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

)+

(p− p∗)δ

(1− δp

n+ ρ

)dΩ.

It is easy to see from Green’s first identity that

I1 ≤ −∫

Ω

d1u

∗|∆n|2

n2+d2v∗|∆p|2

γδp2

dΩ ≤ 0.

Furthermore

I2 =

∫Ω

(n− n∗)2

(αp∗

(1 + αn)(1 + αn∗)(1 + βp∗)+

g

(h+ n∗)(h+ n)− 1

)dΩ

+

∫Ω

(n− n∗)(p− p∗)(

−1

(1 + αn)(1 + βp)(1 + βp∗)+

1

δ(n+ ρ)

)dΩ

−∫

Ω

(p− p∗)2 1

n+ ρdΩ.

The above equation can be rewritten as follows

I2(t) = −∫

Ω

(n− n∗, p− p∗)

(k(n, p) l(n, p)∗ m(n, p)

)(n− n∗p− p∗

)dΩ, (23)

where

k(n, p) = 1− αp∗

(1 + αn)(1 + αn∗)(1 + βp∗)− g

(h+ n∗)(h+ n),

l(n, p) =1

2

1

(1 + αn)(1 + βp)(1 + βp∗)− 1

2

1

δ(n+ ρ),

m(n, p) =1

n+ ρ.


Then it is obvious that dVdt

< 0 if and only if the matrix integrand of (23) is positivedefinite which is equivalent to φ1(n, p) = k +m > 0 and φ2(n, p) = mk − l2 > 0, where

φ1 = 1 +1

n+ ρ− αp∗

(1 + αn)(1 + αn∗)(1 + βp∗)− g

(h+ n∗)(h+ n), (24)

φ2 =1

n+ ρ− αp∗

(n+ ρ)(1 + αn)(1 + αn∗)(1 + βp∗)− g

(n+ ρ)(h+ n∗)(h+ n),

− 1

4(1 + αn)2(1 + βp)2(1 + βp∗)2− 1

4δ2(n+ ρ)2

+1

2

1

(1 + αn)(1 + βp)(1 + βp∗)δ(n+ ρ). (25)

By the conditions (21) and (22), we conclude that φ1(n, p) > 0 and φ2(n, p) > 0. Hencethe positive equilibrium point E∗ is globally asymptotically stable.

4 The existence and non-existence of non-constant

positive equilibria

In this section, our aim is to study the existence and non-existence of non-constant positiveequilibria of steady state problem (26). The corresponding steady-state problem of (4)takes the following form:

−d1∇2n = n

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

), x ∈ Ω,

−d2∇2p = γp

(1− δp

n+ ρ

), x ∈ Ω, (26)

∂n

∂n=∂p

∂n= 0, x ∈ ∂Ω.

4.1 A priori estimate

Now we introduce some lemmas that will be used in this subsection.

Lemma 11 (Maximum Principle). Let g(x, u) ∈ C(Ω × R1) and bj(x) ∈ C(Ω), j =1, 2, · · · , N.

1. If u(x) ∈ C2(Ω)∩C1(Ω) satisfies ∆u(x)+∑N

j=1 bj(x)uxj +g(x, u(x)) ≥ 0 in Ω, ∂u∂n≤

0 and u(x0) = maxΩ u, then g(x0, u(x0)) ≥ 0.

2. If u(x) ∈ C2(Ω)∩C1(Ω) satisfies ∆u(x)+∑N

j=1 bj(x)uxj +g(x, u(x)) ≤ 0 in Ω, ∂u∂n≥

0 and u(x0) = minΩ u, then g(x0, u(x0)) ≤ 0.

Lemma 12 (Harnack Inequality). Let c(x) ∈ C(Ω) and u(x) ∈ C2(Ω) ∩ C1(Ω) be posi-tive solution to ∆u(x) + c(x)u(x) = 0 in Ω subject to homogeneous Neumann boundarycondition. There exists a positive constant C = C(N,Ω, ‖c(x)‖∞) such that maxΩ u(x) ≤C minΩ u(x).


By using the hypothesis of standard regularity theory for elliptic equations discussed in[21], it is obvious that the positive solution of (26) is in C2(Ω) ∩ C1(Ω). Therefore, byusing Lemmas 11 and 12 in (26), we can have some estimates. For convenience, we denoteΛ = Λ(α, β, γ, δ, ρ, g, h) in the sequel.

Theorem 13 (Upper bounds). For any positive solution of (26),

maxx∈Ω

n(x) ≤ 1 and maxx∈Ω

p(x) ≤ 1 + ρ

δ. (27)

Proof. By Theorem 3 and comparison argument to (26), we easily obtain (27).

Theorem 14 (Lower bounds). Let d be a fixed positive constant. Then, for d1, d2 > d,there exists a positive constant C = C(Λ, d) such that

minx∈Ω

n(x) ≥ C and minx∈Ω

p(x) ≥ C. (28)

Proof. Let

n(x1) = minx∈Ω

n(x), n(x2) = maxx∈Ω

n(x), p(y1) = miny∈Ω

p(y), p(y2) = maxy∈Ω

p(y).

By Lemma (11), we have

1 ≤ n(x1) +p(x1)

(1 + αn(x1))(1 + βp(x1))+

g

h+ n(x1), (29)

1 ≤ δp(y1)

n(y1) + ρ, (30)

1 ≥ δp(y2)

n(y2) + ρ. (31)

From (30) and (31), we respectively get

ρ+ n(x1)

δ≤ ρ+ n(y1)

δ≤ p(y1), (32)

andρ+ n(x2)

δ≥ ρ+ n(y2)

δ≥ p(y2). (33)

Since 0 < 1(1+αn(x1))(1+βp(x1))

< 1, it follows from (33) that

1 ≤ n(x1) +p(x1)

(1 + αn(x1))(1 + βp(x1))+

g

h+ n(x1)

≤ n(x1) + p(x1) +g

h

≤ n(x1) + p(y2) +g

h

≤ n(x1) +ρ+ n(x2)

δ+g

h≤ n(x1) +Mn(x2), (34)


where M is a large enough positive constant such that δ−1(ρ+ n(x2)) + gh−1 ≤Mn(x2).Let

c(x) = d−11

(1− n− p

(1 + αn)(1 + βp)− g

h+ n

).

Then, by Lemma 12, it is known that

n(x2) ≤ C1n(x1), (35)

where C1 depends on ‖c(x)‖. From (34), we obtain

minx∈Ω

n(x) = n(x1) ≥ (MC1 + 1)−1 := C2. (36)

It follows from (32) that

minx∈Ω

p(x) = p(y1) ≥ δ−1ρ+ δ−1(MC1 + 1)−1 := C3. (37)

Finally letting C = minC2, C3 completes the proof.

4.2 Non-existence of non-constant positive equilibrium

In this subsection, we study the non-existence of non-constant positive equilibrium (26)under larger diffusion coefficients.

Theorem 15 There exist two constants d∗1, d∗2 and if d1 > d∗1 and d2 > d∗2, then the system

(26) has no non-constant solutions.

Proof. Suppose that (n, p) is a positive solution of (26) and let n =1

Ω

∫Ω

n(x)dx and

p =1

Ω

∫Ω

p(x)dx. By multiplying n− n with first equation of (26), integrating on Ω and

using no-flux boundary condition, we arrive at

d1

∫Ω

|∆(n− n)|2dx =

∫Ω

g1(n, p)(n− n)dx =

∫Ω

(n− n)(g1(n, p)− g1(n, p)dx

=

∫Ω

(n− n)

(n− n2 − uv

(1 + αn)(1 + βp)− gu

h+ n

−n+ n2 +np

(1 + αn)(1 + βp)+

gn

h+ n

)dx

=

∫Ω

(1− n− n− p

(1 + αn)(1 + αn)(1 + βp)

− gh

(h+ n)(h+ n)

)(n− n)2

− n(n− n)(p− p)(1 + αn)(1 + βp)(1 + βp)

dx.


According to Theorems 13 and 14, it follows that

d1

∫Ω

|∆(n− n)|2dx

≤∫

Ω

(1− δC

(1 + α)2(δ + β(1 + ρ))− gh

(h+ 1)2

)(n− n)2dx

−∫

Ω

(δ2C

(1 + α)(δ + β(1 + ρ))2

)(n− n)(p− p)dx, (38)

where C is as defined as in above theorem. Following the same method as above to secondequation of (26), we get the following

d2

∫Ω

|∆(p− p)|2dx

=

∫Ω

g2(n, p)(n− n)dx =

∫Ω

(p− p)(g2(n, p)− g2(n, p)dx

= γ

∫Ω

(p− p)(p− δp2

n+ ρ− p+

δp2

n+ ρ

)dx

= γ

∫Ω

(1− δ(p+ p)

n+ ρ

)(p− p)2 +

δp2(n− n)(p− p)(n+ ρ)(n+ ρ)

dx,

≤∫

Ω

(γ − 2γδC

1 + ρ

)(p− p)2dx+

γ(1 + ρ)2

δ(C + ρ)2(n− n)(p− p)dx. (39)

Thus, by Young’s inequality, we obtain

d1

∫Ω

|∆(n− n)|2dx+ d2

∫Ω

|∆(p− p)|2dx

≤∫

Ω

(1− δC

(1 + α)2(δ + β(1 + ρ))− gh

(h+ 1)2

)(n− n)2dx

+

∫Ω

γ

(1− 2δC

1 + ρ

)(p− p)2dx

+

∫Ω

(γ(1 + ρ)2

δ(C + ρ)2− δ2C

(1 + α)(δ + β(1 + ρ))2

)(n− n)(p− p)dx,

≤∫

Ω

1− δ2C(1 + βC)

(δ + αδ + β(1 + ρ))2− gh

(h+ 1)2+

1

4ε

(γ(1 + ρ)2

δ(C + ρ)2

− δ2C

(1 + α)(δ + β(1 + ρ))2

)(n− n)2dx+

∫Ω

γ − 2γδC

1 + ρ

+ε

(γ(1 + ρ)2

δ(C + ρ)2− δ2C

(1 + α)(δ + β(1 + ρ))2

)(p− p)2dx,

where ε is a small enough positive value. Then, by using Poincar inequality, we obtain∫Ω

(d1µ2|n− n|2 + d2µ|p− p|2)dx ≤∫

Ω

(d1|∆(n− n)|2 + d2|∆(p− p)|2)dx. (40)


Let

d∗1 =1

µ2

1− δ2C(1 + βC)

(δ + αδ + β(1 + ρ))2− gh

(h+ 1)2

+1

4ε

(γ(1 + ρ)2

δ(C + ρ)2− δ2C

(1 + α)(δ + β(1 + ρ))2

)and

d∗2 =1

µ2

γ − 2γδC

1 + ρ+ ε

(γ(1 + ρ)2

δ(C + ρ)2− δ2C

(1 + α)(δ + β(1 + ρ))2

).

If d1 > d∗1 and d2 > d∗2, then we obtain n = n and p = p. Hence the proof is complete.

4.3 Existence of non-constant positive equilibria

In this subsection, we study the existence of non constant positive solutions of (26).From now on, the diffusion coefficients d1, d2 vary while other parameters are kept fixed.Based on Theorem 8, the necessary conditions for no non-constant solution of (26) are1 < αp∗

(1+αn∗)2(1+βp∗)+ g

(h+n∗)2and (h − g)(δ + βρ) < ρh. So we assume these conditions

throughout this section. By Theorems 10 and 13, there exists a constant C and we definethe following set

B(C) :=

(n, p) ∈ X|C−1 < n, p < C. (41)

Using notations as in Section 3.2, (26) can be rewritten as follows

−∆z = D−1G(z), (42)

∂z

∂n= 0. (43)

Thus z is positive solution to (26) if and only if

F (d1, d2, z) := z− (I −∆)−1[D−1G(z) + z] = 0, on X, (44)

where I is the identity operator and (I−∆)−1 is the inverse of (I−∆). Since F is compactperturbation of the identity operator, the Leray-Schauder degree deg(F, 0, B(C)) is well-defined if G 6= 0 on ∂B. Furthermore we note that

DzF (d1, d2, E∗) := I − (I −∆)−1(D−1Gz(E

∗) + I). (45)

We recall that if DzF (d1, d2, E∗) is invertible, the index G(d1, d2, z) at the isolated fixed

point E∗ is defined as index F (d1, d2, E∗) = (−1)r, where r is the number of eigenvalues of

DzF (d1, d2, E∗) with negative real parts. IfG 6= 0 on ∂B(C), then deg(F (d1, d2, z), 0, B(C))

is equal to the sum of the indexes over all isolated solutions to F (d1, d2, z) := 0 in B(C).Using the decomposition (14), we discuss the eigenvalues of DzF (d1, d2, E

∗). Weknow that Wij is invariant under DzF (d1, d2, E

∗) for each integer i ≥ 1 and each integer1 ≤ j ≤ dimE(µi). Thus λ is an eigenvalue of DzF (d1, d2, E

∗) on Wi if and only if it isan eigenvalue of the matrix

I − 1

1 + µiD−1Gz(E

∗) + I =1

1 + µiµiI −D−1Gz(E

∗). (46)


Thus DzF (d1, d2, E∗) is invertible if and only if

I − 1

1 + µiD−1Gz(E

∗) + I (47)

is nonsingular. Thus λ is an eigenvalue of DzF (d1, d2, E∗) on Wi if and only if λ(1 + µi)

is an eigenvalue of Mi where

Mi = µiI −D−1Gz(E∗) =

(µi − d−1

1 g11(E∗) −d−11 g12(E∗)

−d−12 g21(E∗) µi − d−1

2 g22(E∗)

). (48)

Obviously

detMi = d−11 d−1

2 d1d2µ2i − [d2g11(E∗) + d1g22(E∗)]µi + detGz(E

∗). (49)

The trace of Mi istr Mi = 2µi − d−1

1 g11(E∗)− d−12 g22(E∗). (50)

Let H(d1, d2, µ) = d1d2µ2 − [d1g22(E∗) + d2g11(E∗)]µ + detGz(E

∗). Then H(d1, d2, µi) =d1d2 detMi. If

[d1f22(E∗) + d2f11(E∗)]2 > 4d1d2 detGz(E∗), (51)

then H(d1, d2, µ) = 0 has to real solutions:

µ+(d1, d2) =d1g22(E∗) + d2g11(E∗) +

√∆

2d1d2

, (52)

µ−(d1, d2) =d1g22(E∗) + d2g11(E∗)−

√∆

2d1d2

, (53)

where∆ = (d1g22(E∗) + d2g11(E∗))2 − 4d1d2 detGz(E

∗).

Let A(d1, d2) = µ |µ ≥ 0, µ−(d1, d2) < µ < µ+(d1, d2), Sp = µ1, µ2, · · · and m(µi) bethe multiplicity of the eigenvalue µi.

Lemma 16 Suppose H(d1, d2, µi) 6= 0 for all µi ∈ Sp; then

index(G(d,d2, ·.), z∗) = (−1)σ, (54)

where

σ =

∑

µi∈A∩Sp

m(µi), A ∩ Sp 6= ∅,

0, A ∩ Sp = ∅.(55)

In particular, if H(d1, d2, µ) > 0 for any µ > 0, then σ = 0.

,

Theorem 17 Assume 1 < αp∗

(1+αn∗)2(1+βp∗)+ g

(h+n∗)2, (h − g)(δ + βρ) < ρh and f11/d1 ∈

(µn, µn+1) for some n and rn =∑n

1=1 is odd, then there exists d∗ > 0 such that (26) hasat least one non-constant positive solution when d2 ≥ d∗.


Proof. If (10) holds with conditions H1 or H2 or H3 or H4, then (26) has the uniquepositive constant equilibrium E∗. Furthermore, if 1 < αp∗

(1+αn∗)2(1+βp∗)+ g

(h+n∗)2, then

g11(E∗) > 0 and f22(E∗) > 0. It follows that if d2 is large enough, then (51) is easilyobtained and

0 < µ−(d1, d2) < µ+(d1, d2) = 0. (56)

Furthermore

limd2→∞

µ+(d1, d2) = f11/d1, limd2→∞

µ−(d1, d2) = 0. (57)

Since f11/d1 ∈ (µn, µn+1), there exists d0 >> 1 such that

µ+(d1, d2) ∈ (µn, µn+1), 0 < µ−(d1, d2) < µ2, for any d2 ≥ d0. (58)

According to Theorem 15, we know that there exists a large enough d0 such that d1 > d0

and that (26) corresponding to d1 = d, d2 ≥ d has no non-constant positive solution.Moreover we can choose large enough d such that 0 < f11/d1 < µ2, so there exists d∗ > dsuch that

0 < µ−(d1, d2) < µ+(d1, d2) < µ2, for any d2 ≥ d∗. (59)

Now we prove that (26) has at least one non-constant positive solution. The proof isbased on the method of contrary. We assume that this assertion is not true for d2 ≥ d∗.Then a required contradiction is made by using a homotopy argument. Fix d2 = d∗ anddefine

D(t) =

(td1 + (1− t)d 0

0 td2 + (1− t)d∗). (60)

Consider the following problem

−∆z = D−1G(z),∂z

∂n= 0.

(61)

Then z is a non-constant positive solution of (26) if and only if it is a non-constant positivesolution of equation (61) when t = 1. It is clear that E∗ is the unique constant positivesolution of (61) for t ∈ [0, 1]. We know that z is a positive solution of (61) for t ∈ [0, 1] ifand only if

G(t, z) := z− (I −∆)−1[D−1G(z) + z] = 0, (62)

where z ∈ X+. Obviously F (1, z) = F (d1, d2, z), see equation (44). By Theorem 15, it iswell known that F (0, z) = 0 has only the positive constant solution E∗ in X+. By directcomputation, we obtain

DzF (t, E∗) = I − (I −∆)−1(D−1Gz(E∗) + I). (63)

In particular

DzF (0, E∗) = I − (I −∆)−1(D−1Gz(E∗) + I), (64)

DzF (1, E∗) = I − (I −∆)−1(D−1Gz(E∗) + I) = DzF (d1, d2, E

∗), (65)


where D−1 = diagd, d∗. From (58) and (59), we have A(d1, d2) ∩ Sp = ∅ and A(d, d∗) ∩Sp = ∅. Since rn is odd, Lemma 16 yields

index(F (1, ·), E∗) = (−1)rn = −1, index(F (0, ·), E∗) = (−1)0 = −1. (66)

From Theorems 13 and 14, there exist C(d, d1, d∗, d∗2, Λ) > 0 and C(d, d1, d

∗, d∗2, Λ) > 0such that the positive solution of (26) satisfies C < n(x), p(x)C for all t ∈ [0, 1]. DefineB∗ = z ∈ X2, C < n(x), p(x) < C, x ∈ Ω. Then F (t, z) 6= 0 for all z ∈ ∂B∗ andt ∈ [0, 1]. By using homotopy invariance of the Leray-Schauder degree, we get

deg(F (1, ·), B∗, 0) = deg(F (0, ·), B∗, 0). (67)

Note that both equations F (0, z) = 0 and F (1, z) = 0 have unique positive solution n∗ inB∗ and

deg(F (0, ·), B∗, 0) = index(F (0, ·), E∗) = 1, (68)

deg(F (0, ·), B∗, 0) = index(F (1, ·), E∗) = −1, (69)

which contradict (67). The proof is complete.

5 Numerical Simulation

In this subsection, we perform numerical simulations to illustrate our proposed theoreticalresults in previous sections. To do so, we consider the 1D diffusive system with zero fluxboundary condition with diffusion coefficients d1 and d2. Let us take the fixed valuesl = 10, d1 = 1.0, d2 = 0.5, α = 1.9, β = 0.12, δ = 0.3, ρ = 0.2 and γ as bifurcationparameter. Then the model (4) becomes

∂n

∂t= ∇2n+ n

(1− n− p

(1 + 1.9n)(1 + 0.12p)− 0.1

0.8 + n

),

∂p

∂t= 0.5∇2p+ γp

(1− 0.3p

n+ 0.2

),

(70)

for x ∈ (0, 10), t > 0. The above system exhibits a unique positive interior equilibrium(n∗, p∗) = (0.120686, 1.06895). When γ = 0.1, we can get αp∗

(1+αn∗)2(1+βp∗)+ g

(h+n∗)2=

0.252458 < 1. Hence Theorem 8 holds and the interior equilibrium point (n∗, p∗) islocally asymptotically stable which shows in Figure 1 initial condition (n0, p0) = (n∗ +0.1 sin(x), p∗+0.1 sin(x)). Also we get γ∗ = 0.037311. When γ crosses through γ∗, (n∗, p∗)loses its stability and family of periodic solutions occurs (that is, Hopf bifurcation). Figure2 shows the spatially homogeneous periodic of the system (70) γ = 0.02 under initialcondition (n0, p0) = (n∗ + 0.1 sin(x), p∗ + 0.1 sin(x)).

Conclusion

In this paper, we have considered and investigated the dynamics of modified Leslie-Gowerpredator-prey model with Crowley-Martin functional response and nonlinear prey harvest-ing subject to the no-flux boundary condition. The harvesting term has been incorporated


02

46

810

0

500

10000

0.05

0.1

0.15

0.2

0.25

xtime (t)

Pre

y u(

x,t)

02

46

810

0

500

10000.95

1

1.05

1.1

1.15

1.2

1.25

xtime (t)

Pre

dato

r v(

x,t)

Figure 1: Numerical simulations showing the constant steady state of the system (70) isto be locally asymptotically stable when γ = 0.1 and initial condition (n0, p0) = (n∗ +0.1 sin(x), p∗ + 0.1 sin(x)) .

02

46

810

0

500

10000

0.1

0.2

0.3

0.4

xtime (t)

Pre

y u(

x,t)

02

46

810

0

500

10000.9

1

1.1

1.2

1.3

xtime (t)

Pre

dato

r v(

x,t)

Figure 2: Numerical simulations showing a spatially homogeneous periodic of the system(70) is to appear when γ = 0.02 < γ∗ and initial condition is (n0, p0) = (n∗+0.1 sin(x), p∗+0.1 sin(x)) .

into the prey equation to make the model more realistic. We have derived the conditionsfor local stability and Hopf bifurcation of equilibrium points. By using Lyapunov func-tion, the global stability of interior equilibrium point has been analyzed. Also we havediscussed the priori estimate, existence and non-existence of non-constant positive steadystate solutions. Finally numerical simulations have been given to exemplify the efficacyof theoretical results.

References

[1] W. Abid, R. Yafia, M. A. Aziz-Alaoui, H. Bouhafa, and A. Abichou. Global dynam-ics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evol. Equ. Control Theory, 4(2):115-129, 2015.

[2] N. Ali and M Jazar. Global dynamics of a modified Leslie-Gower predator-preymodel with Crowley-Martin functional responses. J. Appl. Math. Comput., 43(1-2):271-293, 2013.

[3] M.A. Aziz-Alaoui and M.D. Okiye. Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math.Lett., 16(7): 1069-1075, 2003.


[4] J. R. Beddington. Mutual interference between parasites or predators and it’s effecton searching efficiency. J. Anim. Ecol., 44(1): 331-340, 1975.

[5] A. A. Berryman. The orgins and evolution of predator-prey theory. Ecology, 73(5):1530-1535, 1992.

[6] Q. Bie, Q. Wang, and Z. Yao. Cross-diffusion induced instability and pattern for-mation for a Holling type-II predator-prey model. Appl. Math. Comput., 247: 1-12,2014.

[7] J. Cao and R. Yuan. Bifurcation analysis in a modified Lesile-Gower model withHolling type II functional response and delay. Nonlinear Dyn., 84(3): 1341-1352,2016.

[8] D. L. DeAngelis, R.A. Goldstein, and R.V. O’neill. A model for tropic interaction.Ecology, 56(4): 881-892, 1975.

[9] C. S. Holling. Some characteristics of simple types of predation and parasitism. Can.Entomol., 91: 385-398, 1959.

[10] D. Indrajaya, A. Suryanto, and A. R. Alghofari. Dynamics of modified Leslie-Gowerpredator-prey model with Beddington-DeAngelis functional response and additiveallee effect. Int. J. Ecol. Dev., 31(3): 60-71, 2016.

[11] P. H. Leslie. Some further notes on the use of matrices in population mathematics.Biometrika, 35(3/4): 213-245, 1948.

[12] P. H. Leslie and J. C. Gower. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika, 47(3/4): 219-234, 1960.

[13] H. Li and Y. Takeuchi. Dynamics of the density dependent predator-prey systemwith Beddington-DeAngelis functional response. J. Math. Anal. Appl., 374(2): 644-654, 2011.

[14] Y. Li and M. Wang. Dynamics of a diffusive predator-prey model with modifiedLeslie-Gower term and Michaelis-Menten type prey harvesting. Acta Appl. Math.,140(1): 147-172, 2015.

[15] Z. Li, M. Han, and F. Chen. Global stability of a stage-structured predator-preymodel with modified Leslie-Gower and Holling-type II schemes. Int. J. Biomath.,5(06): 1250057, 2012.

[16] A. J. Lotka. Elements of Mathematical Biology. Dover New York, 1956.

[17] A. Oaten and W. W. Murdoch. Functional response and stability in predator-preysystems. Am. Nat., 109(967): 289-298, 1975.

[18] P. J. Pal and P. K. Mandal. Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong allee effect.Math. Comput. Simulation, 97: 123-146, 2014.


[19] H. -B. Shi and S. Ruan. Spatial, temporal and spatiotemporal patterns of diffusivepredatorprey models with mutual interference. IMA J. Appl. Math., 80(5): 1534-1568, 2015.

[20] R. Yang and C. Zhang. Dynamics in a diffusive modified Leslie-Gower predator-preymodel with time delay and prey harvesting. Nonlinear Dyn., 87(2): 863-878, 2017.

[21] H. Yin, X. Xiao, X. Wen, and K. Liu. Pattern analysis of a modified Leslie-Gowerpredator-prey model with Crowley-Martin functional response and diffusion. Com-put. Math. Appl., 67(8): 1607-1621, 2014.

[22] S. Yu. Global stability of a modified Leslie-Gower model with Beddington-DeAngelisfunctional response. Adv. Difference Equ., 2014(1): 84, 2014.

[23] R. Yuan, W. Jiang, and Y. Wang. Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting. J. Math. Anal.Appl., 422(2): 1072-1090, 2015.

[24] J. Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model withCrowley-Martin functional responses. Commun. Pure Appl. Anal., 14(3): 1127-1145,2015.

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