Chaos in a Predator-Prey System with Impulsive Perturbations and Stage Structure for the Predator

Abstract

We investigate a predator-prey model with stage structure for the predator and periodic constant impulsive perturbations. Conditions for extinction of prey and immature predator are given. By using the Floquet theory and small amplitude perturbation skills, we consider the local stability of prey, immature predator eradication periodic solution. Furthermore, by using the method of numerical simulation, the influence of the impulsive control strategy on the inherent oscillation is investigated, which shows rich complex dynamic (such as periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows).

1. Introduction

Stage structure model has received much attention in the recent years [1–3]. In this paper, based on the model of [1], we study the dynamic complexities of a predator-prey system with stage structure for the predator and periodic constant impulsive perturbations. The model takes the form

𝑥̇(𝑡)=𝑥(𝑡)(𝑟−𝑎𝑥(𝑡)−𝑏𝑦₂(𝑡)𝑦̇1+𝑚𝑥(𝑡)),₁(𝑡)=𝑘𝑏𝑥(𝑡)𝑦₂(𝑡)1+𝑚𝑥(𝑡)−(𝐷+𝑣₁)𝑦₁𝑦̇(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₂(𝑡)=𝐷𝑦₁(𝑡)−𝑣₂𝑦₂(𝑡),Δ𝑥(𝑡)=−𝛿𝑥(𝑡),Δ𝑦₁(𝑡)=−𝛿₁𝑦₁(𝑡),𝑡=(𝑛+𝑙−1)𝑇,Δ𝑦₂(𝑡)=−𝛿₂𝑦₂(𝑡),Δ𝑥(𝑡)=0,Δ𝑦₁(𝑡)=0,𝑡=𝑛𝑇,Δ𝑦₂(𝑡)=𝑞, (1.1)(1)(2)(3)(4)(5)(6)(7)(8)(9) where 𝑥(𝑡) is the density of prey at time 𝑡,𝑦₁(𝑡),𝑦₂(𝑡) are the densities of immature and mature predators at time 𝑡, respectively. All parameters are positive constants. 𝑏𝑥/(1+𝑚𝑥) is the response function of the mature predator. 𝑟 is the intrinsic growth rate of the prey, 𝑣₁(𝑣₂) is the death rate of the immature (mature) predator, constant 𝑘 denotes the coefficient in converting prey into a new immature predator, and the constant 𝐷 denotes the rate of immature predator becoming mature predator. 0<𝑙≤1, Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡),Δ𝑦₁(𝑡)=𝑦₁(𝑡⁺)−𝑦₁(𝑡),Δ𝑦₂(𝑡)=𝑦₂(𝑡⁺)−𝑦₂(𝑡),and0<𝛿≤1(0<𝛿₁𝛿≤1,0<₂≤1) represents the fraction of prey (immature predator, mature predator) which dies at 𝑡=(𝑛+𝑙−1)𝑇, 𝑇 is the period of the impulsive effect, and 𝑞>0 is the release amount of mature predator at 𝑡=𝑛𝑇.

Recently, it is of great interests to investigate chaotic impulsive differential equations about biological control [4–6]. Gao and Chen [7] and Tang and Chen [8] investigated dynamic complexities in a single-species model with stage structure and birth pulses.

2. Preliminaries

In this section, we give some definitions and lemmas which will be useful to our results.

Let 𝑅₊=[0,∞),𝑅³₊={𝑥∈𝑅³∶𝑥≥0}. Denote by 𝑓=(𝑓₁,𝑓₂,𝑓₃) the map defined by the right hand of the first, second, and third equations of system (1.1). Let 𝑉∶𝑅₊×𝑅³₊→𝑅₊, and let 𝑉 be said to belong to class 𝑉₀ if

(1)𝑉 is continuous at ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×𝑅³₊⋃((𝑛+𝑙−1)𝑇,𝑛𝑇]×𝑅³₊ for each 𝑥∈𝑅³₊,𝑛∈𝑁,lim_{(𝑡,𝑦)→((𝑛+𝑙−1)𝑇}⁺_,𝑥)𝑉(𝑡,𝑦)=𝑉((𝑛+𝑙−1)𝑇⁺,𝑥), and lim_{(𝑡,𝑦)→(𝑛𝑇}⁺_,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇⁺,𝑥) exist;(2)𝑉 is locally Lipschitzian in 𝑥.

Definition 2.1. Let 𝑉∈𝑉₀, then for (𝑡,𝑥)∈((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×𝑅³₊⋃((𝑛+𝑙−1)𝑇,𝑛𝑇]×𝑅³₊, the upper right derivative of 𝑉(𝑡,𝑥) with respect to the impulsive differential system (1.1) is defined as 𝐷⁺𝑉(𝑡,𝑥)=lim_ℎ→0+1supℎ[𝑉(𝑡+ℎ,𝑥+ℎ𝑓(𝑡,𝑥))−𝑉(𝑡,𝑥)]. (2.1)

Lemma 2.2. Let 𝑋(𝑡) be a solution of system (1.1) with 𝑋(0⁺)≥0, then 𝑋(𝑡)≥0 for all 𝑡≥0. And further 𝑋(𝑡)>0,𝑡>0 if 𝑋(0⁺)>0.

Lemma 2.3. Suppose 𝑉∈𝑉₀. Assume that 𝐷⁺𝑉(𝑡,𝑋)≤𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝑙−1)𝑇,𝑉(𝑡,𝑋(𝑡⁺)≤𝜓_𝑛(𝑉(𝑡,𝑥))),𝑡=𝑛𝑇or𝑡=(𝑛+𝑙−1)𝑇, (2.2) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×𝑅₊⋃((𝑛+𝑙−1)𝑇,𝑛𝑇]×𝑅₊ and for 𝑢∈𝑅₊,𝑛∈𝑁, then lim_{(𝑡,𝑣)→(𝑛𝑇}⁺_,𝑢)𝑔(𝑡,𝑣)=𝑔(𝑛𝑇⁺,𝑢) and lim_{(𝑡,𝑣)→((𝑛+𝑙−1)𝑇}⁺_,𝑢)𝑔(𝑡,𝑣)=𝑔((𝑛+𝑙−1)𝑇⁺,𝑢) exist, 𝜓_𝑛∶𝑅⁺→𝑅⁺ is nondecreasing. Let 𝑟(𝑡)=𝑟(𝑡,𝑡₀,𝑢₀) be maximal solution of the scalar impulsive differential equation 𝑢̇(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝑙−1)𝑇,𝑢(𝑡⁺)=𝜓_𝑛(𝑢(𝑡)),𝑡=𝑛𝑇or𝑡=(𝑛+𝑙−1)𝑇,𝑢(0⁺)=𝑢₀, (2.3) existing on [0,∞), then 𝑉(0⁺,𝑥₀)≤𝑢₀, implies that 𝑉(𝑡,𝑥(𝑡))≤𝑟(𝑡),𝑡≥𝑡₀, where 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑢₀) is any solution of system (1.1).

In the following, we give some basic properties about the following subsystem (2.4) and (2.7) of system (1.1):

𝑦̇₂(𝑡)=−𝑣₂𝑦₂(𝑡),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝑙−1)𝑇,Δ𝑦₂(𝑡)=−𝛿₂𝑦₂(𝑡),𝑡=(𝑛+𝑙−1)𝑇,Δ𝑦₂𝑦(𝑡)=𝑞,𝑡=𝑛𝑇,₂(0⁺)=𝑦⁰₂. (2.4) Clearly

𝑦^∗₂(𝑡)=𝑞exp(−𝑣₂(𝑡−(𝑛−1)𝑇))1−(1−𝛿₂)exp(−𝑣₂𝑇),(𝑛−1)𝑇<𝑡≤(𝑛+𝑙−1)𝑇,𝑞(1−𝛿₂)exp(−𝑣₂(𝑡−(𝑛−1)𝑇))1−(1−𝛿₂)exp(−𝑣₂𝑇),(𝑛+𝑙−1)𝑇<𝑡≤𝑛𝑇, (2.5)

(𝑦^∗₂(0⁺)=𝑦^∗₂(𝑛𝑇⁺)=𝑞/(1−(1−𝛿₂)exp(−𝑣₂𝑇)),𝑦^∗₂(𝑙𝑇⁺)=(𝑞(1−𝛿₂)exp(−𝑣₂𝛿𝑙𝑇))/(1−(1−₂)exp(−𝑣₂𝑇))) is a positive periodic solution of system (2.4). Since

𝑦₂(𝑡)=(1−𝛿₂)^(𝑛−1)(𝑦₂(0⁺𝑞)−1−(1−𝛿₂)exp(−𝑣₂𝑇))exp(−𝑣₂𝑡)+𝑦^∗₂(𝑡)(𝑛−1)𝑇<𝑡≤(𝑛+𝑙−1)𝑇,(1−𝛿₂)^𝑛(𝑦₂(0⁺𝑞)−1−(1−𝛿₂)exp(−𝑣₂𝑇))exp(−𝑣₂𝑡)+𝑦^∗₂(𝑡)(𝑛+𝑙−1)𝑇<𝑡≤𝑛𝑇, (2.6) is the solution of system (2.4) with initial value 𝑦⁰₂≥0, where 𝑡∈((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇)⋃((𝑛+𝑙−1)𝑇,𝑛𝑇),𝑛∈𝑁, we get the following.

Lemma 2.4. Let 𝑦^∗₂(𝑡) be a positive periodic of system (2.4), and for every solution 𝑦₂(𝑡) of system (2.4) with 𝑦⁰₂, one has |𝑦₂(𝑡)−𝑦^∗₂(𝑡)|→0, when 𝑡→∞, 𝑦̇₁(𝑡)=−(𝐷+𝑣₁)𝑦₁(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,Δ𝑦₁(𝑡)=−𝛿₁𝑦₁𝑦(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₁(0⁺)=𝑦⁰₁. (2.7)

Lemma 2.5. In subsystem (2.7) exists a equilibrium 𝑦^∗₁(𝑡)=0.

Therefore, we obtain a prey, immature predator eradication periodic solution (0,0,𝑦^∗₂(𝑡)) of system (1.1).

3. Extinction and Boundary

In this section, we study the stability of prey, immature predator eradication periodic solution of system (1.1), and we investigate the boundary of system (1.1).

Theorem 3.1. Let (𝑥(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) be any solution of system (1.1), then (0,0,𝑦^∗₂(𝑡)) is locally asymptotically stable if 𝑟𝑇−𝑏𝑞(1−𝛿₂exp(−𝑣₂𝑙𝑇)−(1−𝛿₂)exp(−𝑣₂𝑇))𝑣₂(1−(1−𝛿₂)exp(−𝑣₂1𝑇))<ln1−𝛿. (3.1)

Proof. The local stability of periodic solution (0,0,𝑦^∗₂(𝑡)) may be determined by considering the behavior of small amplitude perturbations of the solution. Define 𝑥(𝑡)=𝑢(𝑡),𝑦₁(𝑡)=𝑢(𝑡),𝑦₂(𝑡)=𝑤(𝑡)+𝑦^∗₂(𝑡), then 𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)=Φ(𝑡)𝑢(0)𝑣(0)𝑤(0)0≤𝑡<𝑇, (3.2) where Φ(𝑡) satisfies Φ̇(𝑡)=𝑟−𝑏𝑦^∗₂00𝑘𝑏𝑦^∗₂−(𝐷+𝑣₁)00𝐷−𝑣₂Φ(𝑡), (3.3) and Φ(0)=0, the identity matrix. The impulsive perturbations of system (1.1) become 𝑢((𝑛+𝑙−1)𝑇⁺)𝑣((𝑛+𝑙−1)𝑇⁺)𝑤((𝑛+𝑙−1)𝑇⁺)=1−𝛿0001−𝛿₁0001−𝛿₂,𝑢((𝑛+𝑙−1)𝑇)𝑣((𝑛+𝑙−1)𝑇)𝑤((𝑛+𝑙−1)𝑇)𝑢(𝑛𝑇⁺)𝑣(𝑛𝑇⁺)𝑤(𝑛𝑇⁺)=.100010001𝑢(𝑛𝑇)𝑣(𝑛𝑇)𝑤(𝑛𝑇) (3.4) The stability of the periodic solution (0,0,𝑦^∗₂(𝑡)) is determined by the eigenvalues of 𝑀=1−𝛿0001−𝛿₁0001−𝛿₂100010001Φ(𝑇), (3.5) where 𝜇₁=(1−𝛿₁)exp(−(𝐷+𝑣₁)𝑇)<1,𝜇₂=(1−𝛿₂)exp(−𝑣₂𝑇)<1,𝜇₃=(1−𝛿)exp(∫^𝑇₀(𝑟−𝑏𝑦^∗₂(𝑡))𝑑𝑡) according to Floquet theory, the solution (0,0,𝑦^∗₂(𝑡)) is locally stable if |𝜇₃|<1, this is to say, 𝑟𝑇−(𝑏𝑞(1−𝛿₂exp(−𝑣₂𝑙𝑇)−(1−𝛿₂)exp(−𝑣₂𝑇)))/𝑣₂(1−(1−𝛿₂)exp(−𝑣₂𝑇))<ln1/(1−𝛿).

Theorem 3.2. There exists a positive constant 𝑀 such that 𝑥(𝑡)≤𝑀,𝑦₁(𝑡)≤𝑀,𝑦₂(𝑡)≤𝑀 for each solution (𝑥(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of system (1.1) with all 𝑡 large enough.

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) be any solution of system (1.1). Define function 𝑉(𝑡)=𝑉(𝑡,𝑋(𝑡))=𝑘𝑥(𝑡)+𝑦₁(𝑡)+𝑦₂(𝑡), when 𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇, we select 0<𝐿<min{𝑣₁,𝑣₂}, then 𝐷⁺𝑉(𝑡)+𝐿𝑉(𝑡)=𝑘(𝑟+𝐿)𝑥−𝑘𝑎𝑥²+(𝐿−𝑣₁)𝑦₁+(𝐿−𝑣₂)𝑦₂≤𝑘(𝑟+𝐿)𝑥−𝑘𝑎𝑥²≤𝑀₀, (3.6) where 𝑀₀=𝑘(𝑟+𝐿)²/4𝑎. When 𝑡=(𝑛+𝑙−1)𝑇,𝑉((𝑛+𝑙−1)𝑇⁺)≤𝑉((𝑛+𝑙−1)𝑇),𝑡=𝑛𝑇,𝑉(𝑛𝑇⁺)≤𝑉(𝑛𝑇)+𝑞.
According to Lemma 2.3, for 𝑡≥0 we have 𝑉(𝑡)≤𝑉(0)𝑒^−𝐿𝑡+𝑀₀𝑙(1−𝑒^−𝐿𝑡𝑒)+𝑞^{−𝐿(𝑡−𝑇)}1−𝑒^𝐿𝑇𝑒+𝑞^𝐿𝑇𝑒^𝐿𝑇→𝑀−1𝑙⁰𝑒+𝑞^𝐿𝑇𝑒^𝐿𝑇−1(𝑡→∞). (3.7)
Therefore 𝑉(𝑡) is ultimately bounded, and we obtain that each positive solution of system (1.1) is uniformly ultimately bounded.

4. Numerical Analysis

In this section, we study the influence of impulsive period 𝑇 and impulsive perturbation 𝑞 on complexities of the system (1.1).

Let 𝑟=1,𝑎=1,𝑏=4,𝑘=2,𝑚=0.8,𝐷=0.4,𝑣₁=𝑣₂=0.2,𝑙=0.5. Since the corresponding continuous system (1.1) cannot be solved explicitly and system (1.1) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. So we have to study system (1.1) numerically integrated by stroboscopically sampling one of the variables over a range of 𝑇,𝑞 values. The bifurcation diagram provides a summary of essential dynamical behavior of system.

From Theorem 3.1, we know that the prey, immature predator eradication periodic solution (0,0,𝑦^∗₂(𝑡)) is locally stable if 𝑞>𝑞_max=0.7397, which is shown in Figure 1. when 𝑞=0.74, we find that the variable 𝑦₂(𝑡) oscillates in a stable cycle. In contrast, the variables 𝑥(𝑡),𝑦₁(𝑡) rapidly decrease to zero.

(a)

(b)

(c)

(d)

Figure 1 

The dynamics of system (1.1) with 𝑟=1, 𝑎=1, 𝑏=4, 𝑘=2, 𝑚=0.8, 𝐷=0.4, 𝑣1=𝑣2=0.2, 𝑙=0.5, 𝛿=0.1, 𝛿1=0.5, 𝛿2=0.1, 𝑞=0.74, 𝑇=14.5 and initial values 𝑥=8.5, 𝑦1=4.8,𝑦2=2.8. (a) Time series of prey; (b) time series of immature predator; (c) time series of mature predator; (d) phase portrait of system (1.1).

The bifurcation diagram (Figure 2) shows that with 𝑇 increasing from 12.5 to 16, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and nonunique attractors. When 𝑇<𝑇₁≈12.97, system (1.1) has a stable 𝑇-periodic solution. When 𝑇>𝑇₁, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos (Figure 3) (chaotic area with periodic windows). Figure 4 shows the phenomena of non-unique attractors [9]: different attractors can coexist, which one of the attractors is reached depends on the initial values.

(a)

(b)

(c)

Figure 2 

Bifurcation diagrams of system (1.1) showing the effect of 𝑇 with 𝑟=1, 𝑎=1, 𝑏=4, 𝑘=2, 𝑚=0.8, 𝐷=0.4, 𝑣1=𝑣2=0.2, 𝑙=0.5, 𝛿=0.1, 𝛿1=0.5, 𝛿2=0.1, 𝑞=0.5, 12.5≤𝑇≤16 and initial values 𝑥=8.5, 𝑦1=4.8,𝑦2=2.8.

(a)

(b)

(c)

Figure 3 

Period doubling cascade: (a) phase portrait of 𝑇-periodic solution for 𝑇=12.8; (b) phase portrait of 2𝑇-periodic solution for 𝑇=13.5; (c) phase portrait of 4𝑇 for 𝑇=13.8.

(a)

(b)

Figure 4 

Nonunique attractor at 𝑇=13.7: (a) Phase portrait of 3𝑇-periodic solution with initial values 𝑥=8.5,𝑦1=4.8,𝑦2=2.8. (b) Phase portrait of 2𝑇-periodic solution with initial values 𝑥=18.5,𝑦1=14.8,𝑦2=12.8.

(a)

(b)

Figure 5 

Non-unique attractor at 𝑇=15.22: (a) Phase portrait of 3𝑇-periodic solution with initial values 𝑥=8.5,𝑦1=4.8,𝑦2=2.8. (b) Phase portrait of chaotic periodic solution with initial values 𝑥=18.5,𝑦1=14.8,𝑦2=12.8.

The bifurcation diagram (Figure 6) shows that with 𝑞 increasing from 0.01 to 0.8, system (1.1) has rich dynamics including periodic doubling bifurcation, chaos, periodic windows, and periodic halving bifurcation. When 𝑞<𝑞₁≈0.08, system (1.1) has a stable 2𝑇-periodic solution. When 𝑞>𝑞₁, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos with periodic windows (Figure 8) which is followed by a cascade of periodic halving bifurcation from chaos to periodic solution.

(a)

(b)

(c)

Figure 6 

Bifurcation diagrams of system (1.1) showing the effect of 𝑞 with 𝑟=1, 𝑎=1, 𝑏=4, 𝑘=2, 𝑚=0.8, 𝐷=0.4, 𝑣1=𝑣2=0.2, 𝑙=0.5, 𝛿=0.1, 𝛿1=0.5, 𝛿2=0.1, 𝑇=13.7, 0.01≤𝑞≤0.8 and initial values 𝑥=8.5, 𝑦1=4.8,𝑦2=2.8.

(a)

(b)

(c)

Figure 7 

Period doubling cascade: (a) phase portrait of 2𝑇-periodic solution for 𝑞=0.05; (b) phase portrait of 4𝑇-periodic solution for 𝑞=0.08; (c) phase portrait of 4𝑇-periodic solution for 𝑞=0.085.

(a)

(b)

(c)

Figure 8 

Period halving cascade: (a) phase portrait of 𝑇-periodic solution for 𝑞=0.49; (b) phase portrait of 2𝑇-periodic solution for 𝑞=0.52; (c) phase portrait of 𝑇-periodic solution for 𝑞=0.6.

5. Conclusion

In this paper, we investigated the dynamic complexities of a predator-prey system with impulsive perturbations and stage structure for the predator. Conditions for the local asymptotical stability of prey, immature predator eradication periodic solution (0,0,𝑦^∗₂(𝑡)) were given by using the Floquet theory and small amplitude perturbation skills. Using the method of numerical simulation, the influence of impulsive control strategy on inherent oscillation showed that there exists complexity for system (1.1) including periodic doubling bifurcation, periodic halving bifurcation, nonunique attractors, chaos, and periodic windows. All these results showed that dynamical behavior of system (1.1) becomes more complex under periodically impulsive control strategy.