Oscillatory dynamics in a discrete predator-prey model with distributed delays

Changjin Xu; Lilin Chen; Peiluan Li; Ying Guo

doi:10.1371/journal.pone.0208322

Abstract

This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings.

Citation: Xu C, Chen L, Li P, Guo Y (2018) Oscillatory dynamics in a discrete predator-prey model with distributed delays. PLoS ONE 13(12): e0208322. https://doi.org/10.1371/journal.pone.0208322

Editor: Pan-Ping Liu, North University of China, CHINA

Received: March 25, 2018; Accepted: November 15, 2018; Published: December 26, 2018

Copyright: © 2018 Xu et al. 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.

Data Availability: All relevant data are within the manuscript and its Supporting Information file.

Funding: This work is supported by National Natural Science Foundation of China (No.61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020).

Competing interests: The authors have declared that no competing interests exist.

Introduction

It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists [1–5]. In particular, the periodic solutions are of great interest. During the past decades, a great deal of excellent results have been reported for a lot of different continuous or impulsive predator-prey models. For example, Zhang and Hou [6] investigated the four positive periodic solutions of a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms. Liu and Yan [7] considered positive periodic solutions for a neutral delay ratio-dependent predator-prey model with a Holling type II functional response. Liu [8] dealt with the impulsive periodic oscillation of a predator-prey model with Hassell-Varley-Holling functional response. For more related work, one can see [9–27]. Dunkel [28] pointed out that feedback control item in predator-prey models depends on the population number for certain time past and also depends on the average of the population number for a period of time past. In particular, time delay often occur in predator-prey models due to the impact of all the past life history of the predators and preys on their present birth rates. In many cases, the time delay will extend over the entire past due to the intra-species and inter-species competition. Then there is a distribution of delays over a period of time, thus the distributed delays should be incorporated in predator-prey models.

The functional response plays a key role in characterizing the interaction of predators and preys. Based on the experiments of different kinds of species, Holling [29] proposed three types of functional responses: (I) f₁(u) = au, (II) (III) where u(t) represents the prey density at time t, c > 0 is the half-saturation constant, a > 0 denotes the search rate of the predator. Holling type II functional response is most typical of predators that specialized on one or a few prey [29–33]. So in this paper, Holling type II functional response is introduced in model (1).

Motivated by the viewpoint, we proposed the following predator-prey model with Holling II functional response and distributed delays (1) where x_i(t)(i = 1, 2) stands for the prey and predator density at time t, r₁(t) denotes the intrinsic growth rate of prey at time t and r₂(t) denotes the death rate of predator at time t, m > 0 stands for the half-saturation constant, k_i: (−∞, 0] → (0, +∞)(i = 1, 2, 3, 4) is continuous function such that For the biological meaning of model (1), one can see [34].

As pointed out in [35–42], discrete time models are more better to describe the dynamical behaviors than continuous ones since the populations have non-overlapping generations. What’s more, discrete-time systems can provide convenience for numerical simulations. Thus it is interesting to investigate discrete-time systems. The principle aim of this paper is to propose a discrete version of system (1) and analyze the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model.

Discrete version of system (1)

Following [40, 43] and assuming that the average growth rates in system (1) change at regular intervals of time, one has (2) where [t] stands for the integer part of t, t ∈ (0, +∞) and t ≠ 0, 1, 2, ⋯. The solution of (2) possesses the following natures:

is continuous on [0, +∞).
exist for ∀ t ∈ [0, +∞) with the possible exception of the points t ∈ {0, 1, 2, ⋯}, where left-sided derivative exists.
(2) holds ∀ [k, k + 1), where k = 0, 1, 2, ⋯.

Integrating (2) on [k, k + 1), k = 0, 1, 2, ⋯, one has (3) Let t → k + 1, then (3) reads as (4) which is a discrete version of system (1), where k = 0, 1, 2, ⋯.

The following assumptions are made:

(H1) r_i: Z → R⁺ is positive α-periodic (α is a positive integer), i.e., r_i(k + α) = r_i(k)(i = 1, 2), ∀ k ∈ Z.

(H2) The following inequalities hold true.

Existence of positive periodic solutions

First we given two notations: where ℓ(k) is a α–periodic sequence of real numbers defined for k ∈ Z. Let X, Y be normed vector spaces, L: DomL ⊂ X → Y be a linear mapping, N: X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +∞ and ImL be closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P: X → X and Q: Y → Y such that ImP = KerL, ImL = KerQ = Im(I − Q), it follows that L∣DomL ∩ KerP: (I − P)X → ImL is invertible. We denote the inverse of this map by K_P. If Ω is an open bounded subset of X, the mapping N will be called L–compact on if is bounded and is compact. Since ImQ is isomorphic to KerL, there exists a isomorphism J: ImQ → KerL.

Lemma 1 [44] Let L be a Fredholm mapping of index zero and let N be L–compact on If

(a) ∀ ρ ∈ (0, 1), every solution y of Ly = ρNy is such that y ∉ ∂Ω;

(b) QNy ≠ 0, ∀ x ∈ KerL ⋂ ∂Ω, and deg{JQN, Ω ⋂ KerL, 0} ≠ 0, then the equation Ly = Ny has at least one solution lying in

Lemma 2 [40] Let h: Z → R be α periodic, i.e., h(k + α) = h(k), then ∀ ς₁, ς₂ ∈ I_α and ∀ k ∈ Z, one has

Lemma 3 is an α periodic solution of (4) with strictly positive components if and only if is an α periodic solution of (5)

Proof If is an α periodic solution of (4) with strictly positive components, then Hence which leads to (5). If is an α periodic solution of (5), then which leads to (4).

Define

Define |ζ| = max{|ζ₁|, |ζ₂|}, where ζ = (ζ₁, ζ₂)^T ∈ R². Let l^α ⊂ l₂ denote the subspace of all α periodic sequences equipped with the norm ∀ v = {v(k):k ∈ Z} ∈ l^ω. Then l_ω is a finite-dimensional Banach space.

Let (6) (7) then it follows that and are both closed linear subspaces of l^α and

Theorem 1 Let χ₉ be defined by (32). Suppose that (H1), (H2) and hold, then system (4) has at least an α periodic solution with positive components.

Proof. Let X = Y = l^α, (8) (9) where v ∈ X, k ∈ Z and (10) Then L is a bounded linear operator and and then L is a Fredholm mapping of index zero. Define

Then P and Q are continuous projectors such that In addition, K_P: ImL → KerP ⋂ DomL exists and By the equation Lv = ρNv, ρ ∈ (0, 1), one gets (11) Suppose that v(k) = (x₁(k), x₂(k))^T ∈ X is an arbitrary solution of system (11) for a certain ρ ∈ (0, 1) then one has (12) (13) It follows from (11)–(13) that (14) (15) If v = {v(k)} ∈ X, then ∃ ξ_i, η_i ∈ I_α such that (16) By (12) and (13), we have (17) (18) Thus (19) (20) In the sequel, we consider two cases.

(a) If x₁(η₁) ≥ x₂(η₂), then it follows from (12) that which leads to (21) It follows from (19),(21) and Lemma 2 that (22) (23) By (22) and (23), we derive (24) From (13) and (24), we obtain Then (25) Thus by (20), (25) and Lemma 3.2, we get (26) (27) It follows from (26) and (27) that (28)

(b) If x₁(η₁) < x₂(η₂), then it follows from (13) that which leads to (29) It follows from (20),(29) and Lemma 3.2 that (30) (31) By (30) and (31), we derive (32) From (12), we get Then (33) Thus by (19), (33) and Lemma 3.2, we get (34) (35) It follows from (34) and (35) that (36) Then χ_i(i = 1, 2, ⋯, 11) has no relation with ρ ∈ (0, 1). Let M = max{χ₃, χ₆, χ₉, χ₁₂} + M₀, where M₀ > 0 which satisfies , where is the unique positive solution of (5). Thus any solution v = {v(k)} = {(x₁(k), x₂(k))^T} of (11) in X satisfies ‖v‖ < M, k ∈ Z.

Let Ω ≔ {v = {v(k)} ∈ X: ‖v‖ < M}, then Ω is an open, bounded set in X and (a) of Lemma 1 is satisfied. When v ∈ ∂Ω ∩ KerL, v = {(x₁, x₂)^T} with ‖v‖ = max{|x₁|, |x₂|} = M. Then where Let ϕ(x₁, x₂, μ) = μQNv + (1 − μ)Gv, μ ∈ [0, 1], where Letting J be the identity mapping, we have It follows that Lv = Nv has at least one solution in , i.e., (5) has at least one α periodic solution in , say . Let then by Lemma 3 we know that is a α positive periodic solution of system (4). The proof is complete.

Global asymptotic stability

Let the delays be zero, then (4) becomes (37)

Theorem 2 Assume that (H1) and (H2) are satisfied and furthermore suppose that there exist positive constants ν, σ₁ and σ₂ such that (38) Then the positive ω-periodic solution of system (37) is globally asymptotically stable.

Proof In view of Theorem 1, there exists a positive periodic solution of system (37). Make the change of variable (39) It follows from (37) that (40) (41) where converges to zero as ||u|| → 0.

Define a function V by (42) where σ₁ > 0 and σ₂ > 0 are given by (44) and (45) respectively. Calculating the difference of V along the solution of system (40) and (41), we have (43) where (44) (45) It follows from the condition (38) that ∃ ϵ > 0 such that, if k is sufficiently large and ||u|| < ϵ, then (46) In view of Freedman [45], we can see that the trivial solutions of (40) and (41) is uniformly asymptotically stable and so is the solution {(x*(k), y*(k))^T} of (37). The proof is complete.

Remark 1 In [34], Ye et al. investigated the periodic solution of a continuous predator-prey system with Holling type II functional response and infinite delays by applying continuation theorem in coincidence degree theory and some priori estimates on solutions, moreover, this paper does not involve the global asymptotic stability. In this paper, we study the existence of periodic solution of discrete predator-prey model with distributed delays by applying continuation theorem in coincidence degree theory and analyze the global asymptotic stability of periodic solution by Lyapunov function. Form this viewpoint, the results of this article supplement the previous studies of Ye et al. [18].

Numerical example

Example 1 Consider the model as follows: (47) where r₁(k) = 0.6 + sin kπ, r₂(k) = 0.45 + sin kπ, m = 5, k_i(s) = e^s(i = 1, 2, 3, 4). So Thus the conditions (H1)-(H3) of Theorem 3.1 hold true. Therefore, system (47) has at least a positive two-periodic solution (see Figs 1 and 2). Fig 1 shows the changing situation of prey density with the increase of time t; Fig 2 shows the changing situation of predator density with the increase of time t; From Figs 1 and 2, we can see that the prey density and the predator density will keep periodic oscillation with the increase of time t.

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Fig 1. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

https://doi.org/10.1371/journal.pone.0208322.g001

Download:

Fig 2. The relational graph of t, x₁ and x₂.

https://doi.org/10.1371/journal.pone.0208322.g002

Example 2 Consider the model as follows: (48) where r₁(k) = 0.2 + cos kπ, r₂(k) = 0.1 + cos kπ, m = 2, k₁ = 0.2, k₂ = 0.12, k₃ = 0.24, k₄ = 0.35,. So Let σ₁ = 0.18, σ₂ = 0.23, ν = 0.04. Thus the conditions (H1)-(H2) and (38) of Theorem 4.1 are satisfied. Thus the positive two-periodic solution of system (48) is globally asymptotically stable (see Figs 3 and 4). Fig 3 shows the changing situation of prey density with the increase of time t; Fig 4 shows the changing situation of predator density with the increase of time t; From Figs 3 and 4, we can see that the prey density and the predator density will keep globally asymptotically stable periodic oscillation with the increase of time t.

Download:

Fig 3. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

https://doi.org/10.1371/journal.pone.0208322.g003

Download:

Fig 4. The relational graph of t, x₁ and x₂.

https://doi.org/10.1371/journal.pone.0208322.g004

Conclusions

Based on the previous works and some biological meanings of predators and preys, we propose a new discrete delayed predator-prey system. By using the continuation theorem in coincidence degree theory, we present a set of sufficient conditions to ensure to ensure the existence of positive periodic solution of the discrete delayed predator-prey system. In addition, we also discussed the global asymptotic stability of positive periodic solution for the considered system. The obtained theoretical findings have important significance in biological ecology. Considering the effect of random factor, it is meaningful for us to deal with the dynamics of stochastic predator-prey system. This topic will be our future research direction.

Supporting information

S1 Fig. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

https://doi.org/10.1371/journal.pone.0208322.s001

(DOC)

S2 Fig. The relational graph of t, x₁ and x₂.

https://doi.org/10.1371/journal.pone.0208322.s002

(DOC)

S3 Fig. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

https://doi.org/10.1371/journal.pone.0208322.s003

(DOC)

S4 Fig. The relational graph of t, x₁ and x₂.

https://doi.org/10.1371/journal.pone.0208322.s004

(DOC)

Acknowledgments

This work is supported by National Natural Science Foundation of China (No.61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651) and Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020). ORCID iD of the corresponding author: 0000-0002-9306-2061.

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[ref1] 1. Dai BX, Zou JZ. Periodic solutions of a discrete-time nonautonomous predator-prey system with the Beddington-DeAngelis functional response. Journal of Applied Mathematics and Computing. 2007; 24(1-2): 127–139. https://doi.org/10.1007/BF02832305
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[ref3] 3. Zhang Y, Chen SH, Gao SJ, Wei X. Stochastic periodic solution for a perturbed non-autonomous predator-prey model with generalized nonlinear harvesting and impulses. Physica A: Statistical Mechanics and its Applications. 2017; 486: 347–366. https://doi.org/10.1016/j.physa.2017.05.058
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Figures

Abstract

Introduction

Discrete version of system (1)

Existence of positive periodic solutions

Global asymptotic stability

Numerical example

Conclusions

Supporting information

S1 Fig. The time histories of t-x1,t-x2.

S2 Fig. The relational graph of t, x1 and x2.

S3 Fig. The time histories of t-x1,t-x2.

S4 Fig. The relational graph of t, x1 and x2.

Acknowledgments

References

S1 Fig. The time histories of t-x₁,t-x₂.

S2 Fig. The relational graph of t, x₁ and x₂.

S3 Fig. The time histories of t-x₁,t-x₂.

S4 Fig. The relational graph of t, x₁ and x₂.