alexa Analysis of Prey-Predator Model in Chemostat When the Predator Produces Inhibitor

ISSN: 2168-9679

Journal of Applied & Computational Mathematics

Analysis of Prey-Predator Model in Chemostat When the Predator Produces Inhibitor

Moniem AA*
University College in Al-Jamoum, Umm Al-Qura University, Mecca, Saudi Arabia
*Corresponding Author: Moniem AA, University College in Al-Jamoum, Umm Al-Qura University, Mecca, Saudi Arabia, Tel: 966 12 550 1000, Email: [email protected]

Received Date: Jun 15, 2017 / Accepted Date: Oct 03, 2017 / Published Date: Oct 27, 2017

Abstract

In this work, a prey-predator model in chemostat is considered when the predator produces inhibitor. This inhibitor is lethal to the prey by results in decrease of growth rate of the predator at some cost to its reproductive abilities. A Lyapunov function in the study of the global stability of a predator-free steady state is analysed. Local and global stability of other steady states, persistence analysis, as well as numerical simulations are also presented.

Keywords: Persistence; Inhibitor; Chemostat; Prey; Predator; Lyapunov function

Introduction

The chemostat is one of the standard models of an open system in ecology [1-5]. The monograph of Hsu and Waltman has various mathematical methods for analyzing chemostat models [4]. So, it is quite natural that it should be used as a model for studying detoxification problems. Many authors have studied those models [6-12]. Recently, the inhibitor has been introduced in the models for prey - predator in chemostat when the prey produces unaffected inhibitor which is lethal to neither predator nor nutrient [1]. Moniem [2] has considered a model of simple food chain in chemostat when the predator produces unaffected inhibitor which is lethal to neither prey nor nutrient.

In this paper, we consider a prey-predator model in chemo stat when the predator produces inhibitor. This inhibitor is lethal to the prey by results in decrease of growth rate of the predator at some cost to its reproductive abilities.

This paper is organized as follows: In the next section, the model is presented and some simplifications are achieved. Section 3 deals with the existence and local stability of steady states. In section 4, we shall provide global analysis, including global stability of the boundary steady states and persistence analysis. Discussion, comments and numerical simulation are found in final section.

The Model

The interested equations are

Equation (1.1)

Where s(t), x(t), y(t) and p(t) are the concentration of the nutrient, prey, predator and inhibitor at time t respectively. s° denotes the input concentration of the nutrient, D denotes the washout rate, and the parameter γ represents the coefficient of the interaction between the inhibitor and the prey. γi, i=1,2 are the Yield constants. The constant k∈(0,1) represents the fraction of potential growth devoted to producing the inhibitor [3].

Also we have Equation andEquation, where mi, i=1,2 are the maximal growth rates, and ai, i=1,2 are the Michaels- Menten constants.

Now to perform the usual scaling for the chemostat, let

Equation

Substituting in eqn. 1.1 and dropping the bars, the mathematical model will be reduced to the form

Equation (1.2)

Existence and Local Stability

Let T=s+x+y+p then we have Equation. Since each component is non-negative, the system in eqn. 1.2 is dissipative and thus, has a compact, global attractor. To simplify in eqn. 1.2, let Equation, we find that the system in eqn. 1.2 will take the form

Equation (2.1)

Clearly z(t)→0 as t→∞ so the system in eqn. (2.1) may be viewed as an asymptotically autonomous system with the following equations

Equation (2.2)

The equilibrium point E0=(1,0,0) always exists. If 1<f1(1) then there is an equilibrium of in eqn. (2.2) of the form E1=(λs,1-λs,0) where s is the unique solution of f1s)-1=0. Similarly, if Equation, there is an equilibrium of the form E2=(s*λs, y*) where s* is the unique value of s such that Equation is the unique solution of (1-k) f2(x)-1=0 and

Since the limit plane in (eqn. 2.2) is Σ: s+y+z=1 , then by dropping S equation, the system of equations eqn. (2.2) will be reduced to the form

Equation (2.3)

It is easy to show that in eqn. 2.3 in positive plane. As a consequence, the global attractor in eqn. (2.1) lies in the set z=0 and Σ plane where in eqn. (2.3) is satisfied. When the analysis of in eqn. (2.3) is completed in this paper, the work of Thieme (11), relates the corresponding dynamics in eqns. (2.1) and (2.3), and hence in eqn. (1.2). We will show that all solutions in eqn. (2.3) tend to rest points and hence, using Thieme (10), we can find the rest points of the system in eqn. (1.2).

We now discuss the existence of steady state. The washout steady state E0 always exists. A predator-free steady state E1 exists when λS2 exists when SS+λxs)=1-s- λx f1(s) is decreasing function in s with 0<H(0)=1, H(s*)=0 and Equation. If and only if λSx<1.

Next theorem will discuss the local stability of this steady state by finding the eigenvalues of the associated variation matrices.

Theorem 1

If 1<λS then only E0 exists and E0 is locally asymptotically stable. If λSS+λx then only E0 and E1 exist, E0 is unstable, and E1 is locally asymptotically stable. If λSS+λx0, E1, E2 exist, E0, E1 are unstable and E2 is locally asymptotically stable if

Equation

and consequently by an application of the Poincare-Bendixson theorem there is a periodic solution in Σ.

Proof: The Jacobian matrix in eqn. (2.3) is taken the form

Equation

The eigenvalues are on the diagonal and the washout steady state will be locally asymptotically stable if and only if f1-1S.

At (λS,(1-λS),0) the Jacobian matrix becomes

Equation

The two eigenvalues are Equation and Equation Therefore the predator – free steady state is asymptotically stable if and only if Equation

At E2 the Jacobian matrix takes the form

Equation

Then E2 is locally asymptotically stable if the determinant of this matrix is positive and its trace is negative, it means that

Equation

Global Analysis

Theorem 2

For 1<λS and for large t all solutions in eqn. 2.2 tends to E0.

Proof: For1<λS and for large t we get s(t)<1 and f1(1)<1Therefore, the second equation 2.2 gives Equation, which imply Equation The third eqn. (2.2) becomes Equation, which leads to Equation. The first equation (2.2) has a solution s=1+(constant) e-t→1 as t→∞.

Theorem 3

If λS<1 ,1<λSx and for large t then all solutions in eqn. 2.2 tend to E1

Proof

Let

Equation (3.1)

and

Equation (3.2)

Let C(u) be a continuously differentiable function and C′(u) be defined by

Equation (3.3)

Note that C′(u) is linear on [1-λS, λx] We may construct a Lyapunov function as follows:

Equation (3.4)

on the set Ψ ={(s,x,y):0<s+x+y<1} where x=1-λS

Differentiate in eqn. 3.4 with respect to time t, we obtain

Equation (3.5)

The term Equation is non-positive for 0<s<1 and equal zero for s∈[0,1) if and only if s=λS or x=0 Since C′(x) = 0 for Equation and C′(u) ≥ 0 for u ≥ 0, then the term Equation is nonpositive for s∈[0,1).

Define

Equation (3.6)

If Equation then Equation and Equation .

Using the definition of η we find that h(s, x, y) ≤ 0 . If 1-λs<x<λx then all terms of h(s,x,y) are non-positive. If Equation, then C′(x) = β and making use of definition of β and η we find that h(s,x,y) will be non-positive and the second term of V vanishes at y=0 therefore V is non-positive on Ψ.

Let M be the largest invariant subset of φ={(s,x,y)∈Ψ:V=0} such that V=0 at S= λs or x=0 and y=0. More further,V is bounded above, any point of the form (s,0,0) cannot be in the limit set of any solution initiating in the interior of R3+. (λs,x,0)M implies that S=s and from the first equation (2.2 ), we get x=1-λs

Therefore M={E1}. This completes the proof.

Theorem 4

If λs<1 and λsX<1 then the system in eqn. (2.2) is uniformly persistence.

Proof

Let Z1={(s,x,y):s∈[0,1] ,x,y∈(0,1]},

Z2 represents sxplane : 0 ≤ s, x ≤ 1,

Z3 represents syplane : 0 ≤ s, y ≤ 1, and

Z = Z2 ∪ Z3

We want to show that Z is a uniformly strong repeller for Z1 Since E0 and E1 are the only steady states in Z. E0 is saddle in R3and its stable manifold is {( s,0, y ) : 0 ≤ y}. Also, E1 is saddle in R3 and its stable manifold is {( s, x,0) : 0 < x}.Then, they are weak reppelers for Z1.The stable manifold structures of E0 and E1 imply that they are not cyclically chained to each other on the boundary Z. Therefore Z is a uniform strong repeller for Z1 (see proposition in eqn.(1.2) of Thieme [12]).

So, there are ε1>0 and ε2>0. such that Equation and Equation where 1 and ε2 are not depending on the initial values in Z1. By using (Thieme [12]), the first eqn. (2.2) yields that there is ε3>0 such that Equation where 3 is not depending on the initial values in Z1Proof is completed.

Conclusion And Numerical Simulation

In this paper, we consider a prey-predator model in chemostat when the predator produces inhibitor. This inhibitor is lethal to the prey by results in decrease of growth rate of the predator at some cost to its reproductive abilities. We found that the washout steady state is the global attractor, if it is the only steady state and Equation. When the washout and the predator free steady states are the only steady states, we found that E0 is unstable and E1 is locally asymptotically stable. E1is global attractor by constructing a Lyapunov function under condition that Equation. We also showed that E2 exists in the sense that the system is uniformly persistent and E2 is locally asymptotically stable if the determinant of this matrix is positive and its trace is negative, it means that

Equation

We find by numerical simulation that eight iterative examples are presented here to show the influence of increasing the parameter k on the dynamical behaviour. In all examples, parameters values in eqn. (2.2) are as follows [2]:

(s(0),x(0),y(0))=(0.1,0.7,0.8),m1=4.0,m2=5.0,a1=0.6,a2=0.5,γ=0.2

And we deduce that when k ∈[0,0.4] the solution appears to approach a periodic solution. So, E0, E1 and E2 lose their stability (Figures 1-3). Those oscillatory solutions appear to be the results of Hopf bifurcations. The numerical simulation shows that the system in eqn. (2.2) has an attracting limit cycle.

applied-computational-mathematics-the-results-bifurcations-k1

Figure 1: The results of Hopf bifurcations and oscillatory solutions are k=0.1.

applied-computational-mathematics-the-results-bifurcations-k2

Figure 2: The results of Hopf bifurcations and oscillatory solutions are k=0.2

applied-computational-mathematics-the-results-bifurcations-k3

Figure 3: The results of Hopf bifurcations and oscillatory solutions oscillatory solutions are k=0.3.

Also, at k ∈[0.4,6.5] the solution approaches a positive steady state. Both E0 and E1 are unstable and E2 is globally asymptotically stable (Figures 4-7).

applied-computational-mathematics-the-solution-oscillatory-k4

Figure 4: The solution approaches positive steady, unstable, stable states and oscillatory solutions are k=0.4.

applied-computational-mathematics-the-solution-oscillatory-k5

Figure 5: The solution approaches positive steady, unstable, stable states and oscillatory solutions are k=0.5.

applied-computational-mathematics-the-solution-oscillatory-k6

Figure 6: The solution approaches positive steady, unstable, stable states and oscillatory solutions are k=0.6.

applied-computational-mathematics-the-solution-oscillatory-k8

Figure 7: The solution approaches positive steady, unstable, stable states and oscillatory solutions are k=0.8

For k ∈[6.5,1] the solution approaches the predator-free steady state. E0 is unstable and E1 is globally asymptotically stable (Figure 8). All left figures plot in time courses and all right figurers plot the trajectory in (s,x,y) space. We also, deduce that no difference between this paper and Moniem [2] as numerical simulation, even γ changes its value on interval [0,1[for each value of k.

applied-computational-mathematics-the-solution-predator-free-k7

Figure 8: The solution approaches the predator-free steady, unstable, stable states and oscillatory solutions k=0.7

References

Citation: Moniem AA (2017) Analysis of Prey-Predator Model in Chemostat When the Predator Produces Inhibitor. J Appl Computat Math 6: 367. DOI: 10.4172/2168-9679.1000367

Copyright: © 2017 Moniem AA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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