Stability and Hopf bifurcation of a producer-scrounger model with age-structure
- Junhao Wen^{1} and
- Peixuan Weng^{1}Email author
https://doi.org/10.1186/s13662-016-0968-2
© Wen and Weng 2016
Received: 31 March 2016
Accepted: 8 September 2016
Published: 22 September 2016
Abstract
We derive a producer-scrounger model with age-structure in scrounger and looks into its dynamics. Using the methods of eigenvalue analysis and Lyapunov function, we find sufficient and necessary conditions for globally asymptotical stability of extinction equilibrium and scrounger-free equilibrium. A so-called basic reproduction ratio \(R_{0}\) was established to determine whether the scrounger is extinct or uniformly persistent. It is found that if \(R_{0}>1\), the mature time τ does change the dynamical behavior of the model. We confirm that Hopf bifurcation happens if the mature time τ increases.
Keywords
1 Introduction
There are three forms of basic interaction between two species: cooperation, competition, and predator-prey, which has been extensively investigated (e.g., see [1, 2] and the references therein). However, interaction forms between species in nature are complex and diverse. To forage foods, some species not only catch and feed on prey, but also can scrounge foods from others. For example, Bugnyar and Kotrschal [3] found that the free-ranging ravens steal wolves’ food in a game park and the wolves prevent their food from being stolen. This phenomenon is called kleptoparasitism and describes an interaction form between two species, scrounger and producer. Here, scrounger steals food from producer.
General speaking, all individuals always have two stages, immature and mature, and they perform very differently in some aspects. For example, the immature ravens are incapable to fly and have to stay in the nest while the mature ravens undertake the task of searching for food. In mathematics, a model with delay τ (being the maturation period) can be established to describe the population dynamics with age-structure (e.g., see [5–13]). In the current paper, we try to establish a producer-scrounger model with age-structure in scrounger, and study the dynamics and the influence of maturation period in the scrounge interaction.
The paper is organized as follows. In Section 2, a producer-scrounger model with age-structure in scrounger is derived. In Section 3, the basic theory, including existence, uniqueness, positivity and boundedness of solutions for the model are discussed. The stability of equilibriums, uniform persistence, and Hopf bifurcation are investigated in Section 4. Some numerical simulations and concluding discussions are given in the last section.
2 Model derivation
Let \(P(t)\) remain the density of producer at time \(t\ge0\), and positive numbers \(b_{1}\), \(a_{1}\), m, d, and θ be of the same meaning as in system (1.1). Assume that \(S(t)\) and \(I(t)\) express the densities of mature scrounger and immature scrounger, respectively, \(b_{2}\) (\(b_{3}\)) is the natural mortality rate of mature (immature) scrounger, \(a_{2}\) (\(a_{3}\)) is the death rate of mature (immature) scrounger caused by intraspecies competition. Furthermore, assume that there exists neither competition between the immature scrounger and mature scrounger nor intraspecies competition between the immature scroungers (i.e., \(a_{3}=0\)). For simplicity, we assume that the spatial environment is homogeneous, and thus \(m(x)\equiv m\) is a constant.
Suppose that there is a resource of one. If the proportion of this resource ultimately acquired by the producers is \(\frac{d}{S(t)+d}\), and the mature scroungers have resource accounted for \(1-\frac {d}{S(t)+d}=\frac{S(t)}{S(t)+d}\). Considering the producer’s ability to discover food, the number of producers, and the energy transition rate, we can regard \(\theta m\frac{P(t)}{S(t)+d}\) as the birth rate of mature scrounger, and then \(i(t,0)=\theta m\frac{P(t)}{S(t)+d}S(t)\).
Remark 2.2
3 Basic theory of solutions
In this section, we study the existence, uniqueness, positivity, and boundedness of solutions of (2.6).
Denote \(X=C([-\tau,0],\mathbb{R}_{+}^{2})\), \(\mathbb{R}_{+}^{2}=\{ (x,y)^{T}:x\ge0,y\ge0\}\). Then X is a Banach space with norm \(\|\phi \|=\max\{|\phi(\xi)|:\xi\in[-\tau,0]\}\), where \(|\cdot|\) is the standard norm in \(\mathbb{R}^{2}\). For \(\sigma>0\) and \(\phi\in C([-\tau,\sigma),\mathbb{R}^{2})\), define \(\phi_{t}\in X\) as \(\phi _{t}(\xi)=\phi(t+\xi)\) for \(t\in[0,\sigma)\) and \(\xi\in[-\tau,0]\).
Lemma 3.1
(Positivity)
The solution \((P(t),S(t),I(t))^{T}\) of (2.5) with initial function \(\phi\geq\mathbf{0}\) is nonnegative on its existence interval. Furthermore, if \(\phi_{1}(0)>0\), \(\phi_{2}(0)>0\), then \((P(t),S(t))> (0,0)\) as long as the solution \((P(t),S(t))\) exists.
Proof
If \(\phi_{1}(0)>0\) and \(\phi_{2}(0)>0\), then a similar argument leads to \((P(t),S(t))> (0,0)\) on its maximal existence interval. The proof is complete. □
The following lemma will lead to the global existence of solution \((P(t;\phi),S(t;\phi))\).
Lemma 3.2
(Boundedness) The solution \((P(t),S(t))^{T}\) of (2.6) satisfying the initial condition (3.1) is bounded as long as it exists.
Proof
In view of Lemmas 3.1 and 3.2, we can draw the following conclusion.
4 Global dynamical properties
In this section, we look into the stability and Hopf bifurcation of system (2.6). Firstly, we investigate the existence of equilibria.
Lemma 4.1
- (1)
The zero equilibrium (extinction equilibrium) \(E_{0}=(0,0)\) always exists.
- (2)
A unique nonzero boundary equilibrium (scrounger-free equilibrium) \(E_{1}= (\frac{m-b_{1}}{a_{1}},0 ):=(p_{1},0)\) exists if and only if \(m>b_{1}\).
- (3)A unique positive equilibrium (coexistence equilibrium) \(E_{*}=(p^{*},s^{*})\) exists if and only ifwhere \(E_{*}=(p^{*},s^{*})\) satisfies$$ \theta me^{-b_{3}\tau}(m-b_{1})>da_{1}b_{2}, $$In addition, if \(E_{*}\) exists, then \(E_{1}\) also exists.$$-b_{1}-a_{1}p^{*}+m\frac{d}{s^{*}+d}=0, \qquad -b_{2}-a_{2}s^{*}+\theta m\frac {p^{*}}{s^{*}+d}e^{-b_{3}\tau}=0. $$
Proof
4.1 Basic reproduction ratio and stability of boundary equilibria
The following theorem describes the stability of extinction equilibrium \(E_{0}\).
Theorem 4.1
If \(m\le b_{1}\), then the extinction equilibrium \(E_{0}=(0,0)\) is globally asymptotically stable; if \(m>b_{1}\), then the extinction equilibrium \(E_{0}\) is unstable.
Proof
It easy to see that (4.3) has and only has two roots, \(-b_{1}+m\) and \(-b_{2}\). If \(m>b_{1}\), then \(-b_{1}+m\) is positive. Thus, \(E_{0}\) is unstable if \(m>b_{1}\).
Remark 4.1
With \(R_{0}\) defined as before, we see that \(R_{0}>1\) is consistent with the condition for the existence of a unique coexistence equilibrium. Further, \(R_{0}\) is in proportion to the birth rate of producer without mature scrounger’s effect \((\theta m\frac{p_{1}}{d})\), the survival rate for immature scrounger (\(e^{-b_{3}\tau}\)), and the average lifetime of mature scrounger (\(\frac{1}{b_{2}}\)).
Theorem 4.2
Assume that \(m>b_{1}\). If \(R_{0}\le1\), then the scrounger-free equilibrium \(E_{1}=(p_{1},0)\) is globally asymptotically stable; if \(R_{0}>1\), then the scrounger-free equilibrium is unstable.
Proof
4.2 Persistence
Theorem 4.3
Assume that \(R_{0}>1\). Then system (2.6) is uniformly persistent. That is, there exists a positive number ε such that \(\liminf_{t\rightarrow\infty}P(t)>\varepsilon\) and \(\liminf_{t\rightarrow\infty}S(t)>\varepsilon\).
Proof
Denote by \(\omega(\phi)\) the omega limit set of the positive half-orbit \(\gamma^{+}{\phi}=\{\Phi(t)\phi: t\ge0\}\) and \(X_{\partial}=\{\phi\in\partial X_{0}: \Phi(t)\phi\in\partial X_{0},t\ge0\}\).
For all \(\phi\in X_{\partial}\), we have \(\Phi(t)\phi\in\partial X_{0}=X\backslash X_{0}=\{\phi=(\phi_{1},\phi_{2})\in X: \phi_{1}(0)=0\mbox{ or }\phi_{2}(0)=0\}\) for any \(t\ge0\). This implies that either \(P(t)=0\) or \(S(t)=0\) for \(t\ge0\). If \(P(t)=0\) for some \(t_{0}\), then for \(t\ge0\), \(P(t)\equiv0\), and further we have \(\omega(\phi)=E_{0}\). Otherwise, \(P(t)>0\) and \(S(t)\equiv0\) for any \(t\ge0\), and then \(\omega(\phi )=E_{1}\). Thus, \(\bigcup_{\phi\in X_{\partial}}\omega(\phi)\subset M:=\{ E_{0},E_{1}\}\), and no subset of M forms a cycle in \(\partial X_{0}\).
According to [16], Theorem 3, there exists a positive number ε such that \(\liminf_{t\rightarrow\infty }P(t)>\varepsilon\) and \(\liminf_{t\rightarrow\infty }S(t)>\varepsilon\). The proof is complete. □
4.3 Stability of coexistence equilibrium and Hopf bifurcation
Theorem 4.4
- (1)
The coexistence equilibrium \(E_{*}\) is globally asymptotic stable for \(\tau=0\).
- (2)
If \(u_{0}>v_{0}\), then the coexistence equilibrium \(E_{*}\) is locally asymptotic stable for any \(\tau\ge0\).
- (3)
If \(u_{0}< v_{0}\), then the coexistence equilibrium \(E_{*}\) is locally asymptotic stable for \(0\le\tau<\tau_{0}\) and unstable for \(\tau>\tau_{0}\). Furthermore, (2.6) undergoes a Hopf bifurcation at \(E_{*}\) when \(\tau=\tau_{0}\). Here \(u_{0}\), \(v_{0}\), \(\tau_{0}\) are constants defined previously.
Proof
In fact, we could discuss the direction and stability of the Hopf bifurcation by using the normal form theory and the center manifold reduction introduced by Hassard et al. [19]. However, the algorithm is standard, and the application of the result needs complex computations. Therefore, we just do some numerical simulations in the next section instead.
5 Numerical simulations and concluding discussions
In the following, we give some discussion on the influence of maturation period τ in the scrounge interaction.
The existence of producer. The existence of producer is just determined by their ability to discover food (m) and the natural mortality rate \(b_{1}\). However, the mature time τ does not determine the existence of producer.
The existence of scrounger. The basic reproduction ratio \(R_{0}\) determines the existence of scrounger. Note that \(R_{0}=\theta m\frac{p_{1}}{db_{2}}e^{-b_{3}\tau}\) decreases as τ increases, and it is a bounded function of variable τ. We find that (i) if \(\theta m p_{1}< db_{2}\), no matter how long the mature time is, the scrounger finally extincts; (ii) if \(\theta m p_{1}>db_{2}\), shorter mature time is in favor of the existence of scrounger.
Global behaviors for \(R_{0}>1\). In the case \(R_{0}>1\), we have no results of the global stability for \(\tau>0\). But in view of the global convergence to the coexistence equilibrium \(E_{*}\) at \(\tau=0\) and numerical simulations, we found that the mature time τ does change the dynamical behavior of the model. We conjecture that (i) when \(u_{0}>v_{0}\), the coexistence equilibrium is globally asymptotically stable for \(\tau>0\); (ii) when \(u_{0}< v_{0}\), the coexistence equilibrium remains globally asymptotically stable for small \(\tau>0\); (iii) when \(u_{0}< v_{0}\), the positive solutions convergence to a periodic solution for large τ.
Note that we also derived another model (2.7) in Section 2, whereas there exists competition between immature scroungers. The investigation of this model seems challenging. We leave it as an open problem.
If \(g(P,S,d)=\frac{d}{S+d}\), (5.1) becomes (1.1) with homogenous spatial environment.
We can also consider other cases such as \(g(P,S,d)=\frac{d^{2}}{S+d^{2}}\) for different species and such that even g actually depends on the producer P. Furthermore, one can derive a new model that involves a continuous immature age structure. We leave these problems for future research.
Declarations
Acknowledgements
Research is supported by NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426). We are very grateful to the anonymous referee and the editors for a careful reading and helpful suggestions, which led to an improvement of our original manuscript.
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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