Dynamical behaviour of a generalist predator-prey model with free boundary
- Zhi Ling†^{1},
- Lai Zhang†^{1, 2, 3},
- Min Zhu^{1, 4}Email author and
- Malay Banerjee^{5}
Received: 25 May 2017
Accepted: 11 September 2017
Published: 25 September 2017
Abstract
In this paper, we consider a free boundary problem describing the invasion of a generalist predator into a prey population. We analytically derive the conditions guaranteeing the existence and uniqueness of the classical solution by means of the Schauder fixed point theorem, and further study the long-time behaviours of these two species. Finally, we numerically investigate the dynamical behaviour during the early invasion stage. Numerical results show that generalist predators are more likely to succeed in alien invasion by reducing the threshold size of the spatial domain of initial invasion, below which invasion fails.
Keywords
1 Introduction
Alien invasion has frequently been reported to cause detrimental impacts on native ecosystems functions by altering population fitness, triggering extinction and secondary extinction of native species [1]. It has been reported that about 42% of all species in the United States are at risk because of competition with or predation by an alien species [2–4]; in other parts of the world, this figure can be even higher [2, 5].
The invaders can be a specialist feeding on a particular prey population, but it can also be a generalist feeding on multiple food sources, and thus their impacts are generally unexpected. A convenient approach to understand the impact of alien invasion is to develop an appropriate mathematical modelling. In this regards, two approaches have been widely applied. One is to develop ordinary differential equations based on the mean field theory to describe the temporal population dynamics, and the other is to develop reaction-diffusion equations to describe the spatio-temporal population dynamics. While the former approach ignores the spatial aspects of alien invasion, the later suffers from the drawback that alien species can instantaneously spread over the entire spatial domain even if they start to invade in a small area.
To best describe the invasion process, free boundary problems were introduced to biology [6–10]. In fact, free boundary problems have received considerable attention in many fields, such as tumor cure [11] and wound healing [12] in medicine, vapor infiltration of pyrolytic carbon in chemistry [13], and expansion of the area infected by the virus in epidemiology [14]. To the best of our knowledge, it was first introduced to biology by Lin to describe the process of a predator invading a prey population [15]. Since then many mathematical models with free boundary have been developed to research biological population [16–19]. For example, Du and Lin in [17] investigated a diffusive logistic model with a free boundary and proved a spreading-vanishing dichotomy. Wang in [20] studied a diffusive logistic equation with a free boundary and sign-changing coefficient and also derived a spreading-vanishing dichotomy. Additionally, Monobe and Wu in [21] introduced the free boundary into a reaction-diffusion-advection logistic model in heterogeneous environment, and obtained the long-time behaviour of the solution and the asymptotic spreading speed.
While predator-prey models with free boundary have attracted great attention, almost all of the studied models assume that the predator is a specialist, which means that the predator will certainly go to extinction in the absence of the focal prey. In reality, it might be not true since most predators are very likely to have alternative food sources [22, 23]. Predators with multiple food sources are called generalist. A question arises how the invasion of generalist predator into a local system affects the population dynamics in a predator-prey model with free boundary, which remains unexplored.
In this model we assume that the alternative food has no dynamics evolution of its own, which means that the alternative food is fixed in constant amounts, availability is significantly high, and hence unaffected due to consumption (see the references [22, 23]). This is apparently a simplification to reduce the dimension of the system from three to two, and thus allows the use of the theory of Schauder fixed point to study the consequences of the availability of alternative food. However, this simplification is justified for many arthropod predators because they can rely on plant-provided alternative food sources such as pollen or nectar, the availability of which is unlikely to be influenced by the predator’s consumption [24–26]. The paper is structured as follows. In Section 2 we derive the conditions guaranteeing the existence and uniqueness of the classic solution to the model (1). In Section 3, we analyse theoretically the long term behaviour of prey and predator. In Section 4, we perform numerical analysis of the population behaviour. The paper ends with a brief conclusion.
2 Global existence, uniqueness and estimate of the solution
In this section, we prove the local existence and uniqueness results to problem (1) by applying contraction mapping theorem, and then we show the global existence using some suitable estimates.
Theorem 2.1
Proof
It is observed that there exists a time T such that the solution exists in time interval \([0,T]\). Since the global existence theorem depends on a prior estimate with respect to \(h'(t)\), in what follows, we aim to derive a prior estimate for any solution of the problem (1).
Theorem 2.2
Proof
Using the strong maximum principle, we can easily see that \(u>0\) in \([0,T]\times[0,\infty)\) and \(v>0\) in \([0,T]\times[0,h(t))\) when the solution exists.
Theorem 2.2 shows that the free boundary is strictly monotonically increasing with time t, which indicates that the domain invaded by the predator v is gradually expanding with time.
Moreover, the other inequalities in Theorem 2.2, in which \(M_{i}\) is independent of T, imply that we can extend the solution of problem (1) to the global range. The proof of the following theorem is omitted here and the interested reader can refer to that of Theorem 2.3 in [17] or Theorem 2.2 in [28].
Theorem 2.3
Problem (1) admits a unique solution \((u, v; h)\), which exists globally in \([0, \infty)\) with respect to t.
3 Long-time behaviour of \((u, v)\)
To begin with, we present the definitions of spreading or vanishing of the predator population and the related comparison principle.
Definition 3.1
If \(h_{\infty}=\infty\) and \(\liminf_{t\rightarrow+\infty} \Vert v(t,\cdot)\Vert _{C([0,h(t)])}>0\), we call the predator spreading, which means that the predator can survive and spread to the whole domain \([0,\infty)\). While if \(h_{\infty}<\infty\) and \(\liminf_{t\rightarrow+\infty} \Vert v(t, \cdot)\Vert _{C([0,h(t)])}=0\), we call the predator vanishing, which means that it will be maintained in a finite region and finally goes to extinction.
Lemma 3.1
Comparison principle
In fact, \((\overline{u}, \overline{v}, \overline{h})\) is called an upper solution of problem (1). We continue to exhibit the following two lemmas, whose proof can be referred to Lemma 3.3 in [29] and Theorem 4.1 in [27], respectively. We omit the details here.
Lemma 3.2
Lemma 3.3
Suppose that \(a^{-1}< e< b\). If \(h_{\infty}<\infty\), then \(h_{\infty }\leq\Lambda\), where \(\Lambda=\frac{\pi}{2}\sqrt{d/(b-e)}\).
The following theorem presents the sufficient condition for the vanishing of predator species.
Theorem 3.1
Proof
Since \(\lim_{t\rightarrow\infty}\bar{u}(t)=1\), by the comparison principle \(u(t, x) \leq\bar{u}(t)\) for all \(t\in[0,\infty)\) and \(x\in[0,\infty)\), we have \(\limsup_{t\rightarrow\infty}u(t,x) \leq1\) uniformly in \([0,\infty)\).
The following theorem exhibits the sufficient condition for the spreading of predator species.
Theorem 3.2
Proof
This idea of the remainder proof comes from [28]. We will construct the following iteration sequences.
Step 1. The constructions of \(\overline{u}_{1}\) and \(\overline{v}_{1}\).
Step 2. The constructions of \(\underline{u}_{1}\) and \(\underline{v} _{1}\).
Step 3. The constructions of \(\underline{u}_{2}\) and \(\underline{v} _{2}\).
Step 4. The constructions of \(\overline{u}_{2}\) and \(\overline{v}_{2}\).
4 Numerical analysis
In this section, we conduct numerical analysis to understand the role of generality of the predator in the invasion process. The above analytical analysis shows the existence of global solutions to the system (1) and further the conditions under which the invader can either spread or vanish as time advances. While these conditions qualitatively show the long-time behaviour of the prey and predator species, they are unfortunately not sufficiently clear because it is unknown when \(h_{\infty}\) is finite or infinite. Understanding the long-time behaviour is numerically challenging, since the spatial domain is infinitely long. To circumvent this difficulty, we consider a spatial domain of finite range and focus on the invasion process of the early stage, which is thought to play a significant role in determining the ultimate fate of the predator species.
To this aim, we restrict the prey species to living in a spatial domain \(x\in[0, L]\) and \(L= 15\), and impose an homogeneous Neumann boundary condition at \(x = L\). To minimise the impact of the right boundary on the potential results, we consider the invasion stage from \(t=0\) to \(t = 15\), at the end of which the invasive species (predator) is far from reaching the right boundary. We employ the numerical scheme in [30] to perform numerical simulation. Moreover, we assume a relatively slow diffusion process by setting \(d = 1.2\) to ensure that the predator will not reach the right boundary at \(t=15\). For the initial condition of the two species, we assume that, prior to the invasion, the prey species is homogeneously distributed in the spatial domain with an equilibrium density \(u(x,0) = 1\). The predator species starts to invade in a small area with a low density \(u(x) = 0.01\), \(x \in(0,h_{0})\).
5 Conclusion
In summary, we consider an invasion scenario of a generalist predator into prey, which is formulated by a predator-prey model with free boundary. In our model (1), the so-called free boundary \(x=h(t)\) characterises the change of the expanding front for the predator v, as well as the long-time behaviours of the predator v and prey u are focussed on. We conclude that the predator will gradually vanish if the limit of front function \(h(t)\) is finite, while the predator will spread if the limit is infinite under some assumptions, and it further will stabilise to an equilibrium. These analytical findings indicate that the change of invasion region to predator can determine whether it successfully invade or not.
On the other hand, the numerical simulations of model (1) are also carried out. Our numerical results show that a generalist predator is more likely to succeed in invasion than a specialist predator. There is a threshold size of the initial spatial domain for a successful invasion below which invasion can fail. Moreover, we also noticed that there is a time period during which invasive specie increases in density slowly. The existence of such a time period implies that invasion can possibly be unsuccessful if we take into account stochastic effects. We conclude that the more general the invasive species, the more likely to be successful the invasion is.
The free boundary in model (1) describes the one-dimensional environment. With regard to realities of situation, however, we realize the two or three-dimensional case to more match the reality. Naturally, there will be more challenges in mathematical analysis and numerical simulation for multi-dimensional free boundary, and we will pay more attention to these questions in future work.
An interesting extension is to explicitly consider the dynamics of the alternative food and investigate how the competition between two preys affect alien invasion. A promising extension is to make an application of the above results to test how such a kind of model with a free boundary problem can be used to solve real problems. Consideration of invasion from a distant place through the boundary of the concerned domain can be modelled accurately only with the help of spatio-temporal models with free boundary conditions.
Notes
Declarations
Acknowledgements
Zhi Ling and Lai Zhang gratefully acknowledge the financial support by the PRC Grant NSFC 11571301.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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