Permanence of a stage-structured predator-prey system with a class of functional responses

Zhihui Ma; Shufan Wang; Wenting Wang; Zizhen Li

doi:10.1016/j.crvi.2011.08.002

Biological modelling / Biomodélisation

Permanence of a stage-structured predator-prey system with a class of functional responses

Zhihui Ma ^1,² ; Shufan Wang ¹ ; Wenting Wang ¹ ; Zizhen Li ^1,²

¹ School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China
² Key Laboratory of Arid and Grassland Agroecology of Ministry of Education, Lanzhou, 730000, China

Comptes Rendus. Biologies, Volume 334 (2011) no. 12, pp. 851-854.

Résumé

Abstract

A stage-structured predator-prey system incorporating a class of functional responses is presented in this article. By analyzing the system and using the standard comparison theorem, the sufficient conditions are derived for permanence of the system and non-permanence of predators.

Métadonnées

Reçu le : 2009-08-13
Accepté le : 2011-08-02
Publié le : 2011-09-15

PMID

DOI : 10.1016/j.crvi.2011.08.002

Mots clés : Predator-prey system, Stage structure, Generalized functional response, Permanence, Non-permanence

Affiliations des auteurs :

Zhihui Ma ^{1, 2} ; Shufan Wang ¹ ; Wenting Wang ¹ ; Zizhen Li ^{1, 2}

¹ School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China
² Key Laboratory of Arid and Grassland Agroecology of Ministry of Education, Lanzhou, 730000, China

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     author = {Zhihui Ma and Shufan Wang and Wenting Wang and Zizhen Li},
     title = {Permanence of a stage-structured predator-prey system with a class of functional responses},
     journal = {Comptes Rendus. Biologies},
     pages = {851--854},
     publisher = {Elsevier},
     volume = {334},
     number = {12},
     year = {2011},
     doi = {10.1016/j.crvi.2011.08.002},
     language = {en},
}

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Zhihui Ma; Shufan Wang; Wenting Wang; Zizhen Li. Permanence of a stage-structured predator-prey system with a class of functional responses. Comptes Rendus. Biologies, Volume 334 (2011) no. 12, pp. 851-854. doi : 10.1016/j.crvi.2011.08.002. https://comptes-rendus.academie-sciences.fr/biologies/articles/10.1016/j.crvi.2011.08.002/

Version originale du texte intégral

1 Introduction

The past decades have witnessed an enormous interest in predator-prey systems, with most of them focusing on homogeneous populations [1–5]. In fact, no natural population is truly homogeneous, many organisms undergo radical changes in many aspects such as the rates of survival, maturation, reproduction and predation while while they are processing their life history. Therefore, stage-structured predator-prey systems have received much attention recently [6–21].

However, these works [7–21] are largely using constant coefficient systems and have stage structures on only one of the interactive species. The research objectives mainly include the stability and permanence of the system, often with one particular type of functional response. In this article, we propose the following variable coefficient predator-prey system with time delays and stage structures for both interactive species, as well as different functional responses, Specifically, we will consider the condition for the permanence of the two species and the non-permanence condition for predators to become extinct:

\begin{matrix} {\dot{x}}_{1} (t) = r_{1} (t) x_{2} (t) - d_{11} x_{1} (t) - r_{1} (t) e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) \\ {\dot{x}}_{2} (t) = r_{1} (t) e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) - d_{12} x_{2} (t) - b_{1} (t) x_{2}^{2} - c_{1} (t) ϕ (x_{2} (t)) x_{2} (t) y_{2} (t) \\ {\dot{y}}_{1} (t) = r_{2} (t) y_{2} (t) - d_{22} y_{1} (t) - r_{2} (t) e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) \\ {\dot{y}}_{2} (t) = r_{2} (t) e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) - d_{21} y_{2} (t) - b_{2} (t) y_{2}^{2} + c_{2} (t) ϕ (x_{2} (t)) x_{2} (t) y_{2} (t) \end{matrix}

(1)

where x₁(t) and x₂(t) denote the immature and mature densities of the prey at time t, respectively. y₁(t) and y₂(t) denote the immature and mature densities of the predator at time t, respectively. r_i(t), b_i(t), c_i(t) (i = 1, 2) are positive and continuous functions for all t ≥ 0.

The above system assumes that the mature predators only feed on the mature prey population, the immature are produced by the mature populations. The birth rate of prey and predators (r_i(t) > 0; i = 1, 2) is proportional to the existing mature population sizes.

The other parameters have the following biological meanings: c₁(t) denotes the capturing rate of mature predators at time t, and c₂(t)/c₁(t) is the rate of conversion of nutrients from the mature prey into the reproduction of the mature predator; b_i(t) > 0 (i = 1, 2) represents the intraspecific competition rate of mature prey and predators at time t, respectively; d₁₁ and d₁₂ are the death rate of immature and mature population of prey respectively, d₂₂ and d₂₁ are the death rate of immature and mature population of predator respectively; τ_i > 0(i = 1, 2) is the length of time from the birth to maturity of the species i. The term r₁(t)e^{−d₁₁τ₁}x₂(t − τ₁) represents the survived prey population born at time t − τ₁, that is, the transition from the immature to mature prey. The term r₂(t)e^{−d₂₂τ₂}y₂(t − τ₂) denotes the survived predator population born at time t − τ₂, that is, the transition from the immature to mature predator.

The term φ(x₂)x₂, the number of prey captured per predator per unit time, is called the predator functional response and satisfies the following assumptions

0 < ϕ (x_{2}) < L < + \infty, \frac{d}{d x_{2}} (ϕ (x_{2}) x_{2}) \geq 0 (x_{2} > 0) .

(2)

The initial conditions of system (1) are given by

x_{i} (θ) = ϕ_{i} (θ) > 0, y_{i} (θ) = ψ_{i} (θ) > 0, ϕ_{i} (0) > 0, ψ_{i} (0) > 0 (i = 1, 2), θ \in [- τ, 0], τ = \max {τ_{1}, τ_{2}} .

(3)

For the continuity of initial conditions, we require further that

x_{1} (0) = \int_{- τ_{1}}^{0} r_{1} (s) ϕ_{2} (s) d s, y_{1} (0) = \int_{- τ_{2}}^{0} r_{2} (s) ψ_{2} (s) d s .

(4)

Again, we define that

\begin{matrix} r_{k}^{s} = \sup_{t \geq 0} r_{k} (t) > 0, c_{k}^{s} = \sup_{t \geq 0} c_{k} (t) > 0, b_{k}^{s} = \sup_{t \geq 0} b_{k} (t) > 0, \\ r_{k}^{i} = \inf_{t \geq 0} r_{k} (t) > 0, c_{k}^{i} = \inf_{t \geq 0} c_{k} (t) > 0, b_{k}^{i} = \inf_{t \geq 0} b_{k} (t) > 0 (k = 1, 2) . \end{matrix}

(5)

2 Positivity and boundedness

Theorem 2.1

Solutions of system(1)with initial conditions(3) and (4)are positive and bounded for all t ≥ 0.

Proof

First we show x₂(t) > 0 for all t ≥ 0. Otherwise, noticing that x₂(t) = φ₂(t) > 0 for all −τ₁ < t < 0.

Then there exists a t^∗ > 0, such that x₂(t^∗) = 0.

Now denoting t₀ = inf {t > 0 | x₂(t) = 0}.

Then t₀ > 0 and from system (1), we have

\begin{matrix} 0 \leq t_{0} \leq τ_{1} \Rightarrow {\dot{x}}_{2} (t_{0}) = r_{1} (t_{0}) e^{- d_{11} τ_{1}} ϕ_{2} (t_{0} - τ_{1}) > 0 . \\ t_{0} > τ_{1} \Rightarrow {\dot{x}}_{2} (t_{0}) = r_{1} (t_{0}) e^{- d_{11} τ_{1}} x_{2} (t_{0} - τ_{1}) > 0 . \end{matrix}

Hence

{\dot{x}}_{2} (t_{0}) > 0

. By the definition of t₀, we have

{\dot{x}}_{2} (t_{0}) \leq 0

. A contradiction.

Thus x₂(t) > 0 for all t > 0.

Again, considering the following equation

\begin{matrix} \dot{u} (t) = - d_{11} u (t) - r_{1} (t) e^{- d_{11} τ_{1}} u (t - τ_{1}) \\ u (0) = x_{2} (0) \end{matrix}

(6)

We can easily obtain that

u (t) = e^{- d_{11} t} [ϕ_{2} (0) - \int_{0}^{t} r_{1} (s) ϕ_{2} (s) d s] .

and

x_{1} (t) > u (t) (0 \leq t < τ_{1}) .

From the initial conditions (3), we have u(τ₁) = 0, then x₁(t) > u(t) > 0 for 0 ≤ t < τ₁.

By induction, we can show x₁(t) > 0 for all t ≥ 0.

Similarly, we can prove that y₁(t) > 0 and y₂(t) > 0 for all t ≥ 0.

Next, we will prove the boundedness of the solutions of system (1).

Defining the function

V (t) = c_{2}^{s} x_{1} (t) + c_{2}^{s} x_{2} (t) + c_{1}^{i} y_{1} (t) + c_{1}^{i} y_{2} (t) .

The derivative of V(t) along the positive solutions of system (1) is

\begin{matrix} \dot{V} (t) = c_{2}^{s} {\dot{x}}_{1} (t) + c_{2}^{s} {\dot{x}}_{2} (t) + c_{1}^{i} {\dot{y}}_{1} (t) + c_{1}^{i} {\dot{y}}_{2} (t) \\ \leq (c_{2}^{s} r_{1}^{s} - c_{2}^{s} d_{12}) x_{2} (t) - c_{2}^{s} b_{1}^{i} x_{2}^{2} (t) - c_{2}^{s} d_{11} x_{1} (t) + (c_{1}^{i} r_{2}^{s} - c_{1}^{i} d_{21}) y_{2} (t) - c_{1}^{i} b_{2}^{i} y_{2}^{2} (t) - c_{1}^{i} d_{22} y_{1} (t) . \end{matrix}

For a positive constant d ≤ min {d₁₁, d₂₂}, then

\dot{V} (t) + d V (t) \leq c_{2}^{s} (r_{1}^{s} - d_{12} + d) x_{2} (t) - c_{2}^{s} b_{1}^{i} x_{2}^{2} (t) + c_{1}^{i} (r_{2}^{s} - c_{1}^{i} d_{21} + d) y_{2} (t) - c_{1}^{i} b_{2}^{i} y_{2}^{2} (t) .

Hence there exists a positive number M, such that

\dot{V} (t) + d V (t) \leq M

. Then we get

V (t) \leq M d^{- 1} + (V (0) - M d^{- 1}) e^{- d t} .

Therefore, the positive solutions of system (1) are bounded. This completes the Proof. □

3 Permanence and non-permanence

Definition 3.1

If there exist positive constants m_x, M_x, m_y and M_y, such that each solution (x₁(t), x₂(t), y₁(t), y₂(t)) of system (1) satisfies

\begin{matrix} 0 < m_{x} \leq {liminf}_{t \to + \infty} x_{i} (t) \leq {limsup}_{t \to + \infty} x_{i} (t) \leq M_{x} (i = 1, 2), \\ 0 < m_{y} \leq {liminf}_{t \to + \infty} y_{i} (t) \leq {limsup}_{t \to + \infty} y_{i} (t) \leq M_{y} (i = 1, 2) . \end{matrix}

Then system (1) is permanent. Otherwise, it is non-permanent.

Lemma 3.2 [22]

Consider the following equation:

\dot{u} (t) = a u (t - τ) - b u (t) - c u^{2} (t)

where a, b, c > 0;u(t) > 0 for −τ ≤ t ≤ 0, we have

(1). If a > b, then $\lim_{t \to + \infty} u (t) = \frac{a - b}{c}$ ,
(2). If a < b, then lim _t→+∞u(t) = 0.

Theorem 3.3

If the following assumptions hold

\begin{matrix} (H_{1}) . r_{1}^{i} x_{2}^{i} > r_{1}^{s} x_{2}^{s} e^{- d_{11} τ_{1}} > 0 \\ (H_{2}) . r_{2}^{i} y_{2}^{i} > r_{2}^{s} y_{2}^{s} e^{- d_{22} τ_{2}} > 0 \end{matrix}

(7)

where

\begin{matrix} x_{2}^{i} = \frac{r_{1}^{i} e^{- d_{11} τ_{1}} - d_{12} - c_{1}^{s} L y_{2}^{s}}{b_{1}^{s}}, x_{2}^{s} = \frac{r_{1}^{s} e^{- d_{11} τ_{1}} - d_{12}}{b_{1}^{i}}, \\ y_{2}^{i} = \frac{r_{2}^{i} e^{- d_{22}} + c_{2}^{i} ϕ (x_{2}^{i}) x_{2}^{i} - d_{21}}{b_{2}^{s}}, y_{2}^{s} = \frac{r_{2}^{s} e^{- d_{22} τ_{2}} + c_{2}^{s} ϕ (x_{2}^{s}) x_{2}^{s} - d_{21}}{b_{2}^{i}} . \end{matrix}

Then system(1)is permanent.

Proof

Let (x₁(t), x₂(t), y₁(t), y₂(t)) be any positive solution of system (1) for t ≥ 0. Firstly, from the second equation of system (1), we have

\begin{matrix} {\dot{x}}_{2} (t) \leq r_{1} (t) e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) - d_{12} x_{2} (t) - b_{1} (t) x_{2}^{2} \\ \leq r_{1}^{s} e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) - d_{12} x_{2} (t) - b_{1}^{i} x_{2}^{2} (t) . \end{matrix}

By Lemma 3.2 and standard comparison theorem, we get

{limsup}_{t \to + \infty} x_{2} (t) \leq \frac{r_{1}^{s} e^{- d_{11} τ_{1}} - d_{12}}{b_{1}^{i}} ė x_{2}^{s} > 0 .

That is, for any ɛ > 0, there exists a T₁ > 0, for any t > T₁ > 0, such that

x_{2} (t) < x_{2}^{s} + ɛ

Again, from the fourth equation of system (1), there exists a T₂ > T₁ > 0, for any t > T₂, we have

\begin{matrix} {\dot{y}}_{2} (t) \leq r_{2} (t) e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) - d_{21} y_{2} (t) - b_{2} (t) y_{2}^{2} + c_{2} (t) ϕ (x_{2}^{s} + ɛ) (x_{2}^{s} + ɛ) y_{2} (t) \\ \leq r_{2}^{s} e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) - d_{21} y_{2} (t) - b_{2}^{i} y_{2}^{2} + c_{2}^{s} ϕ (x_{2}^{s} + ɛ) (x_{2}^{s} + ɛ) y_{2} (t) . \end{matrix}

We obtain that using Lemma 3.2 and standard comparison theorem

{limsup}_{t \to + \infty} y_{2} (t) \leq \frac{r_{2}^{s} e^{- d_{22} τ_{2}} + c_{2}^{s} ϕ (x_{2}^{s} + ɛ) (x_{2}^{s} + ɛ) - d_{21}}{b_{2}^{i}} .

Since ɛ is sufficiently small, then

{limsup}_{t \to + \infty} y_{2} (t) \leq \frac{r_{2}^{s} e^{- d_{22} τ_{2}} - d_{21} + c_{2}^{s} ϕ (x_{2}^{s}) x_{2}^{s}}{b_{2}^{i}} ė y_{2}^{s} > 0 .

Hence, for this ɛ > 0, there exists a T₃ > T₂ > 0, for any t > T₃ > 0, such that

y_{2} (t) < y_{2}^{s} + ɛ

Now, substituting it into the second equation of system (1), we have

\begin{matrix} {\dot{x}}_{2} (t) \geq r_{1} (t) e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) - d_{12} x_{2} (t) - b_{1} (t) x_{2}^{2} (t) - c_{1} (t) (y_{2}^{s} + ɛ) ϕ (x_{2} (t)) x_{2} (t) \\ \geq r_{1}^{i} e^{- d_{11} τ_{1}} x_{2} (t - τ_{1}) - d_{12} x_{2} (t) - b_{1}^{s} x_{2}^{2} (t) - c_{1}^{s} L (y_{2}^{s} + ɛ) x_{2} (t) . \end{matrix}

By the Lemma 3.2 and the standard comparison theory, noticing that ɛ is sufficiently small, we obtain that

{liminf}_{t \to + \infty} x_{2} (t) \geq \frac{r_{1}^{i} e^{- d_{11} τ_{1}} - d_{12} - c_{1}^{s} L y_{2}^{s}}{b_{1}^{s}} ė x_{2}^{i} > 0 .

That is, for this ɛ > 0, there exists a T₄ > T₃ > 0, for any t > T₄ > 0, such that

x_{2} (t) \geq x_{2}^{i} - ɛ

From the fourth equation of system (1), there exists a T₅ > T₄ > 0, for any t > T₅ > 0, we have

\begin{matrix} {\dot{y}}_{2} (t) \geq r_{2} (t) e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) - d_{21} y_{2} (t) - b_{2} (t) y_{2}^{2} (t) + c_{2} (t) ϕ (x_{2}^{i} - ɛ) (x_{2}^{i} - ɛ) y_{2} (t) \\ \geq r_{2}^{i} e^{- d_{22} τ_{2}} y_{2} (t - τ_{2}) - d_{21} y_{2} (t) - b_{2}^{s} y_{2}^{2} (t) + c_{2}^{i} ϕ (x_{2}^{i} - ɛ) (x_{2}^{i} - ɛ) y_{2} (t) . \end{matrix}

Similarly, we obtain that

{liminf}_{t \to + \infty} y_{2} (t) \geq \frac{r_{2}^{i} e^{- d_{22}} - d_{21} + c_{2}^{i} ϕ (x_{2}^{i}) x_{2}^{i}}{b_{2}^{s}} ė y_{2}^{i} > 0 .

That is, for this ɛ > 0, there exists a T₆ > T₅ > 0, for any t > T₆ > 0, such that

y_{2} (t) \geq y_{2}^{i} - ɛ

Again, from the first equation of system (1), we have

\begin{matrix} {\dot{x}}_{1} (t) \leq r_{1} (t) (x_{2}^{s} + ɛ) - d_{11} x_{1} (t) - r_{1} (t) e^{- d_{11} τ_{1}} (x_{2}^{i} - ɛ) \\ \leq r_{1}^{s} (x_{2}^{s} + ɛ) - d_{11} x_{1} (t) - r_{1}^{i} e^{- d_{11} τ_{1}} (x_{2}^{i} - ɛ) . \end{matrix}

Noticing that ɛ is sufficiently small, we get

{limsup}_{t \to + \infty} x_{1} (t) \leq \frac{r_{1}^{s} x_{2}^{s} - r_{1}^{i} e^{- d_{11} τ_{1}} x_{2}^{i}}{d_{11}} ė x_{1}^{s} > 0 .

According to the third equation of system (1) for any t > T₆, and by Lemma 3.2, we have

{limsup}_{t \to + \infty} y_{1} (t) \leq \frac{r_{2}^{s} y_{2}^{s} - r_{2}^{i} e^{- d_{22} τ_{2}} y_{2}^{i}}{d_{22}} ė y_{1}^{s} > 0 .

By the analogous way, we obtain that

\begin{matrix} {liminf}_{t \to + \infty} x_{1} (t) \geq \frac{r_{1}^{i} x_{2}^{i} - r_{1}^{s} e^{- d_{11} τ_{1}} x_{2}^{s}}{d_{11}} ė x_{1}^{i} > 0 . \\ {liminf}_{t \to + \infty} y_{1} (t) \geq \frac{r_{2}^{i} y_{2}^{i} - r_{2}^{s} e^{- d_{22} τ_{2}} y_{2}^{s}}{d_{22}} ė y_{1}^{i} > 0 . \end{matrix}

Therefore, by Definition 3.1, system (1) is permanent. This completes the proof. □

Theorem 3.4

If (H₁) and the following assumption hold

(H_{3}) . r_{2}^{s} e^{- d_{22} τ_{2}} + c_{2}^{s} ϕ (x_{2}^{s}) x_{2}^{s} < d_{21}

(8)

where

x_{2}^{s} = \frac{r_{1}^{s} e^{- d_{11} τ_{1}} - d_{12}}{b_{1}^{i}} .

Then the prey population is permanent while the predators are non-permanent.

Proof

Let (x₁(t), x₂(t), y₁(t), y₂(t)) be any positive solution of system (1) for t ≥ 0. In order to prove the non-permanence of predators and permanence of prey population, we only show that

\lim_{t \to + \infty} y_{i} (t) = 0 (i = 1, 2) .

If the assumption (H₃) holds, then we obtain that

y_{2}^{s} < 0

and

y_{2}^{i} < 0

From the Proof of Theorem 3.3, there exists a T₇ > T₆ > 0, for any t > T₇, we have

\begin{matrix} {limsup}_{t \to + \infty} y_{2} (t) \leq 0, \\ {liminf}_{t \to + \infty} y_{2} (t) \geq 0 . \end{matrix}

(9)

Hence, we get

\lim_{t \to + \infty} y_{2} (t) = 0 .

Again, substituting it into third equation of the system (1), we have

- d_{22} y_{1} (t) \leq y_{1}^{'} (t) \leq - d_{22} y_{1} (t) .

Integrate it from 0 to t, we get

0 < y_{1} (o) e^{- d_{22} t} \leq y_{1} (t) \leq y_{1} (o) e^{- d_{22} t} .

Thus, we obtain that

\lim_{t \to + \infty} y_{1} (t) = 0 .

Therefore, the prey population is permanent and predators are non-permanent. This completes the proof. □

4 Discussion

In this article, we have considered a stage-structured (for both interactive populations) predator-prey system with a class of functional response incorporating discrete time delay, and obtain the sufficient conditions for the permanence of system (1) and the non-permanence of predators. Our work has provided some valuable suggestions for regulating populations and saving endangered species. We see that the predators only forage on mature prey as food resource. Consequently, when the prey population size is lower than a certain level predators will not sustain. This is especially true for mature predators which are responsible for recruitment. In other words, as long as the population sizes of prey and predators maintain at a certain positive level, the predator-prey interaction will continue.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 30970478,31100306).

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^☆ This work was supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2011-48).

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