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On a new ecoepidemiological model for migratory birds with modified LeslieGower functional schemes
 Kuangang Fan^{1},
 Yan Zhang^{2, 3}Email author and
 Shujing Gao^{3}
https://doi.org/10.1186/s1366201608253
© Fan et al. 2016
Received: 29 November 2015
Accepted: 30 March 2016
Published: 7 April 2016
Abstract
Migratory birds are critical to the prevalence of many epidemic diseases. In this paper, a new two species ecoepidemiological model with disease in the migratory prey is formulated. A modified LeslieGower functional scheme, with saturated incidence and recovery rate are considered in this new model. Through theoretical analysis, a series of conditions are established to ensure the extinction, permanence of the disease, and to keep the system globally attractive. It was observed that if the lower threshold value \(R_{*}>1\), the infective population of the periodic system is permanent, whereas if the upper threshold value \(R^{*}\leq1\), then the disease will go to extinction. Our results also show that predation could be a good choice to control disease and enhance permanence.
Keywords
 migratory birds
 LeslieGower functional response
 saturated incidence and recovery rate
1 Introduction
Nowadays, an important issue in applied mathematics is to study the influence of epidemiological parameters on ecological systems. Since KermacMckendric (1927) first proposed the SIR systems, many attentions have been paid to this field. In 1989, Hadeler and Freedman described a model for predator and prey with parasitic infection [1]. From then on, more and more predatorprey models were proposed and discussed under the frame work of ecoepidemiology; see [2–7] and references therein. The biological significance of these works is that we can see how epidemic diseases affect the interactions of prey and predators and how predators act as biological control to disease transmissions. In nature, migratory birds are responsible for the prevalence of many epidemic diseases, such as WNV, which was introduced in the Middle East by migrating white storks [8], HPAI that broke in Mexico in 1994 and was introduced by some wild migrating birds [9, 10], and so on. However, there are few papers analyzing the role of migratory birds, especially by mathematical models and analysis, except the works of Chatterjee et al. [10–13], Gao et al. [14] and Zhang et al. [15].
In [10], Chatterjee and Chattopadhyay assumed the prey population migrated with disease and proposed a oneseason ecoepidemiological predatorprey model for migratory birds. In [11], Chatterjee et al. modified and analyzed their model in [10] by taking time lags into consideration. Their analysis showed that we could control the outbreak of the disease by making use of the time lag factor suitably. In [12], the author introduced standard incidence into the model and obtained the stability of equilibrium point in the presence or absence of environmental fluctuations. Chatterjee (in [13]) discussed an ecoepidemiological model with a nonautonomous recruitment rate and a general functional response. They showed that the contact rate, the predation, and the recovery rate were central to the extinction of the disease. In [14], Gao et al. considered a competitive model for migratory birds and economical birds population. They analyzed the model and discussed dynamics of the model. Zhang et al. in [15] proposed a timedependent model for migratory birds with saturated incidence rate. They also analyzed the dynamics of the system, such as permanence, extinction, and global attractivity of the model. In [15], for simplicity, only the bilinear predation rate was considered for migratory birds and the diversity of the functional responses was not referred to.
As we all know, the functional response is a critical factor in the research of the population dynamics for predatorprey models. The mutual interference between predator and prey can influence the relationship between them. In the past decades, more and more different forms of ratiodependent functional responses were proposed, such as of the CrowleyMartin type, the BeddingtonDeAngelis type, the LeslieGower type, the HassellVarley type, and so on [16]. In this paper, we consider a modified LeslieGower functional response, in which the LeslieGower term is \(\frac{P(t)}{k_{i}(t)+S(t)+I(t)}\), \(i=1,2\), to describe the dynamics between migratory preys and their predators.

\(r(t)\) is the growth rate of the predator population.

\(c_{1}(t)\) (\(c_{2}(t)\)) is the maximum value of per capita rate of S (respectively, I) due to P at time t. Because the predators catch the infected prey more easily than the healthy ones, we have \(c_{1}(t)\leq c_{2}(t)\).

\(c_{3}(t)\) is the maximum value of the per capita rate of P due to S and I at time t ([16]).

\(w_{1}(t)\) denotes the level of environment protection to prey at time t and \(w_{2}(t)\) has a similar meaning to \(w_{1}(t)\).

\(\sigma(t)\) denotes the effects on the predator by absorbing the susceptible prey and \(\sigma(t)\leq1\) for all \(t\geq0\).
The rest of this paper is organized as follows. In Section 2, we analyze the nonautonomous differential equations for migratory birds and establish a set of sufficient conditions to discuss the extinction, the permanence of the disease, and keep the system globally attractive. In Section 3, some results are presented for the periodic system. In Section 4, we verify our theoretical results and outline a discussion by making comparison among the new model (1.2), the SI model (1.1) and the model in [15] with the help of numerical simulation. Finally, some conclusions are given in Section 5.
2 The analysis of the model
To proceed, we give some appropriate definitions and notations and list them in the following.
 (B1)
\(\Lambda(t)\), \(\beta(t)\), \(\gamma(t)\), \(\alpha(t)\), \(d(t)\), \(e(t)\), \(f(t)\), \(\sigma (t)\), \(r(t)\), \(w_{i}(t)\) (\(i=1,2\)) and \(c_{i}(t)\) (\(i=1,2,3\)) are all nonnegative, continuous functions and bounded on \(R_{+}\);
 (B2)there are constants \(\omega_{i}>0\) (\(i=1,2,3,4,5,6\)) satisfying$$\begin{aligned} &\liminf_{t\rightarrow +\infty} \int_{t}^{t+\omega_{1}}\Lambda(\theta)\,d\theta>0,\qquad {\liminf _{t\rightarrow+\infty}} \int_{t}^{t+\omega_{2}}\,d(\theta)\,d\theta>0, \\ & \liminf _{t\rightarrow+\infty} \int_{t}^{t+\omega_{3}}r(\theta)\,d\theta>0, \qquad \liminf_{t\rightarrow+\infty} \int_{t}^{t+\omega_{4}}e(\theta)\,d\theta>0, \\ & \liminf _{t\rightarrow +\infty} \int_{t}^{t+\omega_{5}}\frac{c_{1}(\theta)}{w_{1}(\theta )}\,d\theta>0,\qquad { \liminf_{t\rightarrow+\infty}} \int_{t}^{t+\omega_{6}}\frac {c_{3}(\theta)}{w_{2}(\theta)}\,d\theta>0; \end{aligned} $$
 (B3)
\(d^{m}>0\), \(w_{1}^{m}>0\), \(w_{2}^{m}>0\).
Theorem 2.1
Proof
Theorem 2.2
Proof
Second, we claim that it is impossible that \(I(t)\leq \alpha_{0}\), for all \(t\geq t_{0} \). From this claim, we have two cases. In the first case, there exists a \(T\geq T^{*}\), such that \(I(t)\geq\alpha_{0}\) for all \(t\geq T\) and in the second case, \(I(t)\) oscillates about \(\alpha_{0}\) for all large t.
Next we turn to a discussion of how to control the disease and have the following result.
Theorem 2.3
 (B4)
\(\liminf_{t\rightarrow\infty}\int_{t}^{t+\xi}\beta (\theta)\,d\theta>0\),
 (B5)
\(\limsup_{t\rightarrow +\infty}\frac{1}{\lambda^{*}} \int_{t}^{t+\lambda^{*}} (\frac{\beta(\theta)S_{0}^{*}(\theta )}{1+\gamma(\theta)S_{0}^{*}(\theta)}e(\theta)\frac{f(\theta )}{1+\alpha(\theta)S_{0}^{*}(\theta)} \frac{c_{2}(\theta)p_{0}^{*}(\theta)}{w_{1}(\theta)+S_{0}^{*}(\theta )} )\,d\theta\leq0\),
Proof
First of all, we prove that there is a constant \(t_{1}\geq T\) satisfying \(I(t_{1})<\sigma\), where σ is a sufficiently small positive constant.
Finally, as σ is an arbitrarily small constant, we can obtain \(I(t)\rightarrow0\), as \(t\rightarrow+\infty\).
This completes the proof. □
Next, the global attractivity of the model will be discussed. First, the definition will be given below.
Definition 2.1
([20])
Theorem 2.4
Proof
Remark 1
3 Some results for the periodic system
 (A1)
Parameters \(\Lambda(t)\), \(\beta(t)\), \(\gamma(t)\), \(\alpha(t)\), \(d(t)\), \(e(t)\), \(f(t)\), \(r(t)\), \(\sigma(t)\), \(w_{i}(t)\) (\(i=1,2\)), and \(c_{i}(t)\) (\(i=1,2,3\)) are all nonnegative, continuous periodic functions which have a period \(\omega>0\),
 (A2)
\(\overline{\Lambda}>0\), \(\overline{d}>0\), \(\overline{r}>0\), \(\overline{e}>0\), \(\overline{c_{1}/w_{1}}>0\), \(\overline {c_{3}/w_{2}}>0\),
 (A4)
\(\overline{\beta}>0\).
Corollary 3.1
Corollary 3.2
Corollary 3.3
Corollary 3.4
Remark 2
 (D1)
Parameters \(\Lambda(t)\), \(\beta(t)\), \(\alpha(t)\), \(\gamma(t)\), \(d(t)\), \(e(t)\), \(f(t)\) are all nonnegative, continuous periodic functions which have a period \(\omega>0\),
 (D2)
\(\overline{\Lambda}>0\), \(\overline{d}>0\), \(\overline{e}>0\),
 (D4)
\(\overline{\beta}>0\).
 (1)
If \(\widehat{R}=\frac{\overline{\beta S_{0}^{*}/1+\gamma S_{0}^{*}}}{(\overline{e}+\overline{f/1+\alpha S_{0}^{*}})}\leq1 \), then the infective prey population of model (1.1) goes to extinction;
 (2)
If \(\widehat{R}=\frac{\overline{\beta S_{0}^{*}/1+\gamma S_{0}^{*}}}{(\overline{e}+\overline{f/1+\alpha S_{0}^{*}})}>1 \), then the infective prey population of model (1.1) is permanent.
4 Numerical simulation and discussion
In this section, a set of numerical simulations are carried out to confirm and visualize our theoretical results. The role of predation on the system dynamics is discussed by comparing system (1.2) with the SI model (1.1). Moreover, the effects of the functional response in controlling disease is compared between system (1.2) and the model in [15].
Third, we will study the role of predation on system dynamics through making a comparison between model (1.2) and (1.1).
5 Conclusion
In this paper, a new nonautonomous predatorprey model for migratory birds has been considered. The main results for permanence, extinction of the disease, and global attractivity of the system are obtained in Theorems 2.12.4. Theorem 2.1 shows that the predator and prey in the model are permanent if the condition (2.1), which is the inferior limit of the minimum loss of the predator on interval \([t, t+\omega_{7}]\) for some constant \(\omega_{7}>0\), is established.
In Theorem 2.2, \(s_{0}(t)\) is the density of the susceptible prey without infected prey at time t, satisfying \(\dot{S}(t)=\Lambda(t)\frac{c_{1}(t)}{k_{1}(t)}M_{0}^{2}d(t)S\). It is shown that \(s_{0}(t)\) is a globally attractive state of the susceptible prey. In addition, \(p_{0}(t)\) is the density of the predator without any infected prey at time t, satisfying \(\dot{p}(t)=p (r(t)+\frac{c_{3}(t)}{k_{2}^{2}(t)}\sigma (t)M_{0}^{2}\frac{c_{3}(t)}{k_{2}(t)}p )\). From Lemma 1 of [19], it can be shown that \(p_{0}(t)\) is also a globally attractive state of the predator. Then \(\beta(t)S_{0}(t)e(t)f(t) \frac{c_{2}(t)p_{0}(t)}{k_{1}(t)+S_{0}(t)}\) is the available minimum growth rate of the infected prey at time t. Thus, the left hand of inequality (2.6) implies an inferior limit of the available minimum growth rate of the infected prey in the mean on the interval \([t, t + \lambda]\). By Theorem 2.2, the infected prey will be permanent when the inferior limit is positive.
Theorem 2.3 implies that the infected prey will be extinct when the superior limit of the available maximum growth rate of the infected prey in the mean on interval \([t, t+\lambda^{*}]\) for some constant \(\lambda^{*}>0\) is nonpositive.
In Theorem 2.4, through constructing a Liapunov function, a diagonal dominance condition for the global attractivity of system (1.2) is presented.
Declarations
Acknowledgements
The research have been supported by The Natural Science Foundation of China (11261004), the bidding project of Gannan Normal university (15zb01), The Foundation of Education Committee of Jiangxi (GJJ150674) and the key projects of the Natural Science Foundation of Jiangxi University of Science and Technology (NSFJ2015K09).
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.
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