Dynamic behaviors of a modified predator-prey model with state-dependent impulsive effects
- Shulin Sun^{1}Email author,
- Cuihua Guo^{2} and
- Chao Qin^{1}
https://doi.org/10.1186/s13662-015-0735-9
© Sun et al. 2015
Received: 23 May 2015
Accepted: 23 December 2015
Published: 15 February 2016
Abstract
In this paper, we formulate a two-dimensional autonomous predator-prey model with state-dependent impulsive effects and square root response function. The square root response function indicates that the prey population gather together for self-defense purposes. Firstly, we prove that the system has semitrivial periodic solution under some conditions. Furthermore, by the Poincaré map and the theory of impulsive differential equations, the existence and stability of positive order-1 or order-2 periodic solution of the system are investigated in detail. The validity of all results is illustrated by numerical simulations.
Keywords
state-dependent impulse predator-prey system periodic solution square root response function1 Introduction
Many evolutionary processes in nature are subjected to short temporary perturbations and experience abrupt changes at certain moments of time. The duration of the changes is very short and negligible in comparison with the duration of the process considered. These short-time perturbations are often assumed to be momentary changes or impulses. Therefore, it is realistic that impulsive differential equations model such processes. In the recent ten years, the theory of impulsive differential equations has been extensively used to model many real processes in biology, physics, chemistry, engineering, and other sciences. Particularly, some impulsive differential equations have been introduced successfully in population dynamics (such as fishing or agriculture) and epidemic dynamics; see [1–9] and references therein. In most of the cases, scholars investigate the population dynamical systems with impulsive perturbation at fixed times. However, in practical ecological systems, the implementations of some control strategy (by catching, poisoning or releasing the natural enemy, etc.) depend on the state of target species, which is a more realistic project. This is known as impulsive state feedback control strategy, which is widely used in many biological systems. Recently, a few studies on state-dependent impulse effects were made in [10–18]. In particular, Jiang et al. [10, 11], Nie et al. [12–14] and Zhang et al. [18] investigated some predator-prey systems by using the Poincaré map and theory of impulsive differential equations, the sufficient conditions of existence and stability of semitrivial solution and positive periodic solution were obtained. Guo and Chen [16] and Tian et al. [17] studied the existence and stability of the positive period-1 solution of the system with impulsive state feedback control. The abundant dynamic behaviors of the systems were obtained.
These predator-prey systems with prey group defense ability have more abundant and interesting dynamic characteristics and attract attention of many scholars. A recent novel contribution models the fact that it is the individuals at the edge of the herd that generally suffer the heaviest consequences of the predators’ attacks. Recently, some predator-prey models [19–23] in which the prey exhibits herd behavior were considered. In these models, the predator interacts with the prey along the outer corridor of the herd of prey. As a mathematical consequence of the herd behavior, the interaction terms in systems use the square root of the prey population rather than simply the prey population. The use of the square root properly accounts for the assumption that the interactions occur along the boundary of the population. For example, for drifting herbivores in the savannas, moving in very large herds and subject to individual attacks of predators, the likelihood that they are hunted in the way we describe here is evident [23].
2 Preliminaries
- (H)\(c>s\), where$$ \left \{ \textstyle\begin{array}{@{}l} x^{*}=\frac{s^{2}}{c^{2}},\\ y^{*}=\frac{s(c^{2}-s^{2})}{c^{3}}. \end{array}\displaystyle \right . $$(2.1)
By the biological background of system (1.2) we only consider system (1.2) in the region \(D=R^{2}_{+}\). Obviously, the global existence and uniqueness of solutions of system (1.2) are guaranteed by the smoothness properties of f, which is the mapping defined by the right-hand side of system (1.1); for details, see [24, 25].
In the rest of this paper, we will use the following definitions.
Definition 2.1
[25]
- (1)
Orbitally stable if \(\forall\varepsilon>0\), \(\exists \delta>0\) such that for any \(z^{*}(t_{0})=(x_{0}^{*}, y_{0}^{*})\,\bar{\in}\,\{h\} \times[y^{*}, +\infty)\) satisfying \(\| z^{*}(t_{0})-z(t_{0})\|<\delta\), we have \(\mathrm{d}(z(t; t_{0}, z^{*}(t_{0})), O^{+}(z_{0}, t_{0}))<\varepsilon\) for \(t>t_{0}\).
- (2)
Orbitally attractive if \(\forall\varepsilon>0\), \(\exists\delta >0\) and \(\tilde{T}>0\) such that for any \(z^{*}(t_{0})=(x_{0}^{*}, y_{0}^{*})\,\bar{\in}\, \{h\}\times[y^{*}, +\infty)\) satisfying \(\| z^{*}(t_{0})-z(t_{0})\|<\delta\), we have \(\mathrm{d}(z(t; t_{0},z^{*}(t_{0})), O^{+}(z_{0}, t_{0}))<\varepsilon\) for any \(t>t_{0}+\tilde{T}\).
Lemma 2.1
(Analogue of Poincaré’s criterion)
Now, let \(z(t)=(x(t), y(t))\) be a solution of system (1.2) with initial conditions \(z_{0}=z(t_{0})=((1-p)h, y_{0})\in R_{+}^{2}\). This trajectory \(O^{+}(z_{0},t_{0})\) begins from the point \(E_{0}((1-p)h, y_{0})\) and moves along the solution curve \(z(t)\), then it first intersects the section \(\sum^{h}\) at the point \(F_{0}(h, \widetilde{y}_{0})\), and, next, the point \(F_{0}\) is transferred to the point \(E_{1}((1-p)h, y_{1})\) on the section \(\sum^{p}\) due to the impulse effects, then reaches the point \(F_{1}(h, \widetilde{y}_{1})\) on the section \(\sum^{h}\) again, etc. So, we have two-point sequences \(\{E_{k}((1-p)h, y_{k})\}\) and \(\{F_{k}(h, \widetilde{y}_{k})\}\) (\(k=0,1,2,\ldots\)). In addition, notice that the coordinates satisfy the relation \(y_{k}=(1+q)\widetilde{y}_{k-1}+\alpha\) (\(k=1,2,\ldots\)).
Definition 2.2
A trajectory \(O^{+}(z_{0},t_{0})\) of system (1.2) is said to be order-k periodic if there exists a positive integer \(k\geq1\) such that k is the smallest integer for \(y_{0}=y_{k}\).
3 Main results
3.1 Existence of semitrivial periodic solution with \(\alpha=0\)
3.2 Existence and stability of positive periodic solutions
In this subsection, we give sufficient conditions for the existence and stability of positive periodic solutions in the cases \(h \leq x^{*}\) and \(h > x^{*}\), respectively.
Case I: \(h\leq x^{*}\).
On the existence of positive periodic solution of system (1.2), we have the following theorem.
Theorem 3.1
For any \(q>-1\) and \(\alpha> 0\), system (1.2) has a positive order-1 periodic solution.
Proof
On the other hand, suppose that the vertical isocline \(L: 1 - x - \frac{y}{\sqrt{x}}=0\) intersects the section \(\sum^{p}\) at the point \(E_{0}((1- p)h, (1 -(1-p)h)\sqrt{(1-p)h})\). The trajectory \(O^{+}(E_{0}, t_{0})\) from the initial point \(E_{0}\) intersects the section \(\sum^{h}\) at the point \(F_{1}(h, y_{1})\) with \(y_{1}< y^{*}\), then suddenly jumps to the point \(F_{1}^{+}((1-p)h, (1+q)y_{1}+ \alpha)\) on the section \(\sum^{p}\), and finally reaches the point \(F_{2}(h, y_{2})\) on the section \(\sum^{h}\) again. Suppose that there is a positive constant \(q^{*}\) such that \((1+q)y_{1}+ \alpha=(1-(1-p)h)\sqrt{(1-p)h}\). Then, the point \(F_{1}^{+}\) coincides with the point \(E_{0}\) for \(q=q^{*}\), the point \(F_{1}^{+}\) is above the point \(E_{0}\) for \(q>q^{*}\), and \(F_{1}^{+}\) is under the point \(E_{0}\) for \(q< q^{*}\). However, on account of the point \(E_{0}\) on the isocline L and the phase portrait of system (1.2), we find that the point \(F_{2}\) is under the point \(F_{1}\) for any \(q \in (- 1, q^{*}) \cup (q^{*}, \infty)\).
- (i)
if \(y_{1} = y_{2}\), then system (1.2) has a positive order-1 periodic solution;
- (ii)if \(y_{1} > y_{2}\), then$$ y_{1} -P_{2}(q, \alpha, y_{1}) = y_{1} - y_{2} > 0. $$(3.3)
By (3.2) and (3.3) it follows that the Poincaré map (2.2) has a fixed point, that is, system (1.2) has a positive order-1 periodic solution. This completes the proof. □
Next, we state and prove the stability of the positive order-1 periodic solutions of system (1.2).
Theorem 3.2
Proof
Therefore, by Lemma 2.1, provided that condition (3.4) is satisfied, the order-1 periodic solution \((\phi(t), \psi(t))\) of system (1.2) is orbitally asymptotically stable. This completes the proof. □
Case II: \(h>x^{*}\).
On the existence and stability of positive periodic solution of system (1.2), we have the following theorem.
Theorem 3.3
For \(h>x^{*}\), there is a positive constant \(\alpha^{*}=\alpha (h)>0\) such that for any \(q>-1\) and \(\alpha>\alpha^{*}\), system (1.2) only has a orbitally asymptotically stable positive order-1 or order-2 periodic solution.
Proof
Now, suppose that the trajectory of system (1.2) that starts from the point \(E_{1}((1-p)h,y)\) (\(y\in[\tilde{y}_{1}, \infty)\)) intersects with \(\sum^{h}\) at the point \(E_{2}(h, y_{0})\) for the fist time; then \(y_{0}\in(0, (1-h)\sqrt{h}]\). If \(y_{0}=(1-h)\sqrt{h}\) and \(y=(1+q)y_{0}+\alpha\), then system (1.2) has a positive order-1 periodic solution. Otherwise, from the Poincaré map (2.2) of the section \(\sum^{h}\) it follows that \(y_{1}=P_{2}(q, \alpha,y_{0})\) and \(y_{2}=P_{2}(q, \alpha, y_{1})\); and so forth, we have \(y_{n+1}=P_{2}(q,\alpha,y_{n}) \) (\(n=2,3,\ldots\)). In particular, if \(y_{0}=y_{1}\), then system (1.2) has a positive order-1 periodic solution, and if \(y_{0}\neq y_{1}\) and \(y_{0}=y_{2}\), then system (1.2) has a positive order-2 periodic solution.
Next, similarly to [12, 13, 16], we discuss the general circumstance, that is, \(y_{0}\neq y_{1} \neq y_{2}\neq\cdots\neq y_{n}\) (\(n>2\)).
- (i)
\(y_{2}< y_{0}< y_{1}\).
If \(y_{2}< y_{0}< y_{1}\), then \(y_{3}>y_{1}>y_{0}>y_{2}\) by (3.7). Repeating the process, we have$$0< \cdots< y_{2n} < \cdots< y_{2} < y_{0} < y_{1} < \cdots< y_{2n+1} < \cdots< (1-h)\sqrt{h}. $$ - (ii)
\(y_{0}< y_{2}< y_{1}\).
If \(y_{0}< y_{2}< y_{1}\), then similarly to (i) we have$$y_{0}< y_{2} < \cdots< y_{2n} < \cdots< y_{2n+1} < \cdots< y_{3}< y_{1} < (1-h) \sqrt{h}. $$
- (i)
\(y_{1}< y_{0}< y_{2}\).
If \(y_{1}< y_{0}< y_{2}\), then \(y_{2}> y_{0}> y_{1}> y_{3}\) by (3.7). Repeating the process, we have$$0< \cdots< y_{2n+1}< \cdots< y_{1}< y_{0}< y_{2}< \cdots< y_{2n}< \cdots< (1-h)\sqrt{h}. $$ - (ii)
\(y_{1}< y_{2}< y_{0}\).
If \(y_{1}< y_{2}< y_{0}\), then similarly to (i) we have$$0< y_{1}< \cdots< y_{2n+1}< \cdots< y_{2n}< \cdots< y_{2}< y_{0}< (1-h)\sqrt{h}. $$
From this analysis, in case (i) of (a), it follows from the monotone bounded theorem that \(\lim_{n\rightarrow\infty} y_{2n}=\theta_{2}\) and \(\lim_{n\rightarrow\infty} y_{2n+1}=\theta_{1}\); in addition, \(0< \theta_{2}< y_{0}< \theta_{1}< (r-h)\sqrt{h}\). Thus, we get \(\theta_{2}=P_{2}(q, \alpha, \theta_{1})\) and \(\theta_{1}=P_{2}(q, \alpha, \theta_{2})\). This means that system (1.2) has an orbitally asymptotically stable positive order-2 periodic solution. Similarly, in case (i) of (b), system (1.2) has also an orbitally asymptotically stable positive order-2 periodic solution. In cases (ii) of (a) and (ii) of (b), it follows from the Poincaré map and the closed interval theorem that system (1.2) has an orbitally asymptotically stable positive order-1 periodic solution. The proof is completed. □
Remark
From the proof of Theorem 3.3 we note that the trajectory of system (1.2) passing through the point \(((1-p)h, y)\) (\(y\in(\tilde{y}_{2}, \tilde{y}_{1})\)) will not intersect with \(\sum^{h}\) as time increases and will tend to the focus \((x^{*}, y^{*})\). Therefore, \(\alpha>\widetilde{y_{1}}+ (1-h)\sqrt{h}\) is a sufficient condition ensuring that the trajectory of system (1.2) intersects with \(\sum^{h}\) infinitely many times in view of the impulse effects.
4 Numerical simulation
We have obtained analytical results on dynamical behaviors of a predator-prey model with state-dependent impulse effects in front sections. Now we will illustrate the validity of these results through numerical simulation.
In system (1.1), let \(s=0.5\) and \(c=0.6\). Then the positive equilibrium point \((x^{*}, y^{*})=(0.6945, 0.2546)\) is globally asymptotically stable.
- (1)
\(s=0.5\), \(c=0.6\), \(p=0.2\), \(\alpha=0\), \(h=0.6< x^{*}\), \(q=0.001\).
- (2)
\(s=0.5\), \(c=0.6\), \(p=0.2\), \(\alpha=0\), \(h=0.6< x^{*}\), \(q=0.1\).
- (3)
\(s=0.5\), \(c=0.6\), \(p=0.2\), \(\alpha=0\), \(h=0.72> x^{*}\), \(q=0.1\).
- (1)
\(s=0.5\), \(c=0.6\), \(p=0.2\), \(\alpha=0.2\), \(h=0.6< x^{*}\), \(q=0.1\).
- (2)
\(s=0.5\), \(c=0.6\), \(p=0.2\), \(\alpha=0.45\), \(h=0.72> x^{*}\), \(q=0.1\).
5 Conclusion
On the basis of the predator-prey model (1.1) with square root functional responses, we formulate a new model (1.2) with state impulsive control strategy. In spite of the simplicity of the model, which is composed of two ordinary differential equations together with relations defining the impulsive condition, it is very significant and has interesting dynamical behaviors.
For the predator-prey model (1.1), the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd, and this behavior makes their predators difficult to get food (such as lions on the vast grassland). If this situation continues for a long time in this way, then the predator will be in peril of extinction (see Figure 2). In order to prevent the predator from dying out, we must take some control strategy to ensure that this result does not happen. In this paper, we investigate theoretically the state impulsive control strategy of protecting the predator. Furthermore, we have obtained some interesting results (see Figures 3, 4, 5, 6), which show that h, q, and α are important control parameters. Unfortunately, we cannot currently prove the stability of the semitrivial periodic solution (3.1).
Declarations
Acknowledgements
The authors would like to thank the Journal editors and the anonymous reviewers for their valuable and insightful suggestions that contributed to a significant improvement of the paper. This work is supported by the Natural Science Foundation of Shanxi Province (2013011002-2).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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