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On a new ecoepidemiological model for migratory birds with modified LeslieGower functional schemes
Advances in Difference Equations volume 2016, Article number: 97 (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.
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.
To construct the model for migratory birds, we suppose that the prey population, including the susceptible population, S, and the infected population, I, migrate into the system. The incidence rate and recovery rate are assumed to take saturated forms which are more realistic as many researchers suggested, that is, \(\frac{\beta(t)\mathit{SI}}{1+\gamma(t)S}\) and \(\frac{f(t)}{1+\alpha(t)I}\). Then without predation, the SI model can be expressed as follows:
where \(\Lambda(t)\), \(\beta(t)\) denote the instantaneous recruitment rate of the prey population and the force of the infective (contact rate) at time t. \(f(t)\) represents the recovery rate of the infected prey from the disease. \(d(t)\), \(e(t)\) denote the natural death rate and the mortality rate including the natural death rate and the diseased death rate for susceptible and infective prey population at time t, respectively. Obviously, \(d(t)\leq e(t)\) for all \(t\geq0\). \(\gamma(t)>0\) and \(\alpha(t)>0\) measure the force of the inhibition effect at time t.
We assume that the predator population P eat both the susceptible and the infective prey in a form of modified LeslieGower scheme. Inspired by the above factors, we propose a nonautonomous differential equation for migratory birds,
and the initial conditions are
Denote the set \(\Theta=\{(S,I,P)\in R^{3}:S>0, I>0, P>0\}\), then we can prove that it is a positively invariant set of system (1.2). In fact, letting the right equations of system (1.2) by \(F_{i}(S, I, P)\) (\(i=1, 2, 3\)) and \(X=(S, I, P)^{T} \in R^{3}\), then system (1.2) can be rewritten in a vector form as \(F(X)=[F_{1}(X), F_{2}(X), F_{3}(X)]^{T}\), where \(F \in C^{\infty}(R^{3})\). Thus, system (1.2) becomes
with \(X(0)=X_{0}\in R^{3}_{+}\). Therefore, for any \(X(0)\in R^{3}_{+}\) satisfying \(X_{i}=0\), then \(F_{i}(X)_{X_{i}=0}\geq0\) (\(i=1, 2, 3\)). Thus, the set Θ is positively invariant. (For more details, please see [17].)
The other parameters for model (1.2) are defined as shown below:

\(r(t)\) 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.
The analysis of the model
To proceed, we give some appropriate definitions and notations and list them in the following.
For convenience, we denote
where \(f(t)\) is a continuous and bounded function that defined on \(R_{+}=[0,+\infty)\). Moreover, we make some assumptions as below:

(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
Under assumptions (B1)(B3), if there is a constant \(\omega_{7}>0\) satisfying
where the constant \(M_{1}=\max\{(\frac{\Lambda}{d})^{u}, 1\}\), then both the prey population and the predator population are permanent.
Proof
First of all, suppose that \((S,I,P)\) is an arbitrary positive solution of model (1.2) with initial conditions (1.3). By the first two equations of (1.2), we have
for all \(t\geq0\). Then applying the conclusion of Lemma 2.1 in [18] and the comparison theorem, there are constants \(M_{1}=\max\{(\frac{\Lambda}{d})^{u}, 1\}\) and \(T_{1}>0\) satisfying
Applying (2.2) to system (1.2), we obtain
for all \(t\geq T_{1}\). Using the condition (2.1), Lemma 1 in [19], and the comparison theorem, there are constants \(M_{2}=\max\{(\frac{w_{2}r+M_{1}r}{c_{3}})^{u}, 1\}\) and \(T_{2}\ (\geq T_{1})\) satisfying
Second, from inequality (2.3) and system (1.2), we have
Applying Lemma 2.1 in [18] again, there are constants \(m_{1}=\min\{(\frac{\Lambda}{e+M_{2}(c_{1}+c_{2})/w_{1}})^{v},1\}\) and \(T_{3}>T_{2}\) satisfying
Next, considering the last equation of model (1.2)
Applying Lemma 1 in [19] and the comparison theorem, we see that there exist constants \(m_{2}=\min\{ (\frac{w_{2}r}{c_{3}} )^{v}, 1\}\) and \(T_{4}>T_{3}\) satisfying
Thus, by (2.2)(2.5), we have the following results:
It completes the proof. □
Now we give the results about the permanence of the infective prey. Suppose \(S_{0}(t)\), \(p_{0}(t)\) are an arbitrary fixed solution of the system
and
respectively, where \(M_{0}=\frac{1}{m_{2}}+M_{1}+M_{2}\), then we can obtain the theorem about the permanence of the infective prey population as follows.
Theorem 2.2
Under assumptions (B1), (B2), (B3), if there exist constants \(\lambda>0\) and \(\omega_{8}>0\) satisfying
and
then the infected prey population I is permanent.
Proof
Our proof is motivated by the work of Zhang and Teng [18] and Niu [20]. Choose an arbitrary solution of system (1.2) and denote it by \((S(t),I(t),P(t))\). Then, by (2.2)(2.6), there are constants \(0<\varepsilon_{1}\), \(\varepsilon_{2}<1\), and \(t_{1}>0 \) satisfying
and
for all \(t\geq t_{1}\).
First of all, we prove that there is a constant \(\alpha>0\), being independent of any positive solution of system (1.2) and satisfying
Consider the auxiliary equation
Applying Lemma 4 in [21], we see that for the given constants \(\varepsilon_{1} >0\) and \(M_{0}>0\), there are positive constants \(\delta_{1}=\delta _{1}(\varepsilon_{1})>0\), \(G_{1}=G_{1}(\varepsilon_{1},M_{0})>0\), satisfying for any \(t_{0}\in R_{+}\) and \(x_{0}\in[0,M_{0}] \), if \(\beta(t)M_{0}\alpha< \delta_{1}\) for all \(t\geq t_{0}\),
Here, \(x(t,t_{0},x_{0})\) is the solution of equation (2.10) with initial value \(x(t_{0})=x_{0}\).
In addition, we consider the equation
Based on Lemma 2 in [20], for the given constants \(\varepsilon_{1} >0\) and \(M_{0}>0\), there are positive constants \(\delta_{2}=\delta_{2}(\varepsilon_{1})>0\), \(G_{2}=G_{2}(\varepsilon_{1},M_{0})>0\), satisfying that, for any \(t_{0}\in R_{+}\) and \(M_{0}^{1}\leq v_{0}\leq M_{0} \), if \(\frac{c_{3}(t)}{w_{2}^{2}(t)}M_{0}\alpha< \delta_{2}\) for all \(t\geq t_{0}\), we have
and here, \(v(t,t_{0},v_{0})\) is the solution of equation (2.12) with initial value \(v(t_{0})=v_{0}\).
Choose a constant \(\alpha_{0}=\frac{1}{2} \{\frac{\delta_{1}}{\beta^{u}M_{0}+1},\frac {\delta_{2}}{(c_{3}/w_{2}^{2})^{u} M_{0}+1} \}\) and suppose (2.9) is not true, then for the positive solution \((S(t),I(t),P(t))\) of system (1.2), there exists a \(Z\in R_{3}^{+} \) satisfying initial condition \((S(0),I(0),P(0))=Z\) and
Thus, from the definition of a superior limit, we see that there is a constant \(t_{2}\ (>t_{1})\) such that
for all \(t\geq t_{2}\). Hence, from model (1.2), we obtain
Let \(x(t)\), \(v(t)\) be the solution of equations (2.10), (2.12), which satisfy the conditions \(x(t_{2})=S(t_{2})\) and \(v(t_{2})=P(t_{2}) \), respectively. Applying the comparison theorem, we have
for all \(t\geq t_{2}\). So by (2.11), (2.13) we get
and
Then, from the equation for \(I(t)\) in system (1.2), we further have
where \(T^{*}=t_{2}+G_{1}+G_{2}\), thus
Therefore, from (2.7), we have \(I(t)\rightarrow+\infty\), as \(t\rightarrow+\infty\), which contradicts (2.14). Hence, (2.9) is true.
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.
Obviously, we merely have to take the second case into consideration. Now, we are in a position to prove \(I(t)\geq \alpha_{0} \exp((h_{1}H+h_{2}\lambda))\triangleq m\) for sufficiently large t, where
and
Let \(t_{1}^{*}\), \(t_{2}^{*}\) be sufficiently large such that
If \(t_{2}^{*}t_{1}^{*}\leq H\), then considering the second equation of model (1.2) and integrating it from \(t_{1}^{*}\) to t, we have
If \(t_{2}^{*}t_{1}^{*}> H\), taking a similar proof as that in (2.15), (2.16), we obtain
Then for any \(t\in[t_{1}^{*}, t_{2}^{*}]\), when \(t\leq t_{1}^{*}+H\), we have
When \(t>t_{1}^{*}+H\), we choose a nonnegative integer q such that \(t\in[t_{1}^{*}+H+q\lambda, t_{1}^{*}+H+ (q+1)\lambda)\), then by (2.7), (2.17), and (2.18) we have
Thus, we finally obtain
This completes the proof. □
Next we turn to a discussion of how to control the disease and have the following result.
Theorem 2.3
Under assumptions (B1), (B2), (B3), if there are constants \(\xi,\lambda^{*}>0\) satisfying

(B4)
\(\liminf_{t\rightarrow\infty}\int_{t}^{t+\xi}\beta (\theta)\,d\theta>0\),

(B5)
\(\limsup_{t\rightarrow +\infty}\frac{1}{\lambda^{*}} \int_{t}^{t+\lambda^{*}} (\frac{\beta(\theta)S_{0}^{*}(\theta )}{1+\gamma(\theta)S_{0}^{*}(\theta)}e(\theta)\frac{f(\theta )}{1+\alpha(\theta)S_{0}^{*}(\theta)} \frac{c_{2}(\theta)p_{0}^{*}(\theta)}{w_{1}(\theta)+S_{0}^{*}(\theta )} )\,d\theta\leq0\),
where \(S_{0}^{*}(t)\), \(p_{0}^{*}(t)\) are fixed solutions of the following equations:
and
respectively, then the infected prey I will go to extinction.
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.
By assumption (B4), there are constants \(\eta>0\) and \(T_{0}>0\) satisfying
For any sufficiently small \(0<\sigma<1\), let \(\sigma_{0}=\min\{\frac{\lambda^{*}\eta\sigma}{2\xi},\frac{1}{2}\eta \sigma\}\). If (B5) holds, we can see that there exist \(\delta>0\) and \(T_{1}\geq T_{0}\) satisfying
for all \(t\geq T_{1}\). Let \(n_{0}\) be an integer such that \(\frac{2\xi}{\lambda^{*}}\leq n_{0}\leq\frac{2\xi}{\lambda^{*}}+1\) and \(\lambda_{0}=n_{0}\lambda^{*}\), then
By the first two equations of system (1.2), Lemma 2.1 in [18], and applying the comparison theorem, we see that there is a constant \(T_{2}\geq T_{1}\) satisfying
Moreover, from model (1.2), we also see that there is a \(T_{3}\geq T_{2}\) satisfying
Denote \(T=\max\{T_{2},T_{3}\}\), \(h=\sup_{t\geq T}\{\beta(t)(S_{0}^{*}(t)+\delta)+e(t)+f(t)+\frac {c_{2}(t)p_{0}^{*}(t)}{w_{1}(t)+S_{0}^{*}(t)}+\beta(t)\}\), so for all \(t\geq T\), it yields
Suppose that \(I(t)\geq\sigma\) for all \(t\geq T\), then let \(q^{*}\geq0 \) be an integer satisfying \(t\in[T+q^{*}\lambda_{0}, T+(q^{*}+1)\lambda_{0})\), and we integrate (2.20) from T to t, yielding
Thus, \(I(t)\rightarrow0\) as \(t\rightarrow+\infty\), which contradicts \(I(t)\geq\sigma\), and we can see that there must be a \(t_{1}\geq T\) such that \(I(t_{1})<\sigma\).
Next, we prove that
for all \(t\geq t_{1}\). If the above inequality is not true, then there is a \(t_{2}> t_{1}\) satisfying \(I(t_{2})> \sigma\exp (h\lambda_{0})\). Therefore, there must be a constant \(t_{3}\in(t_{1}, t_{2})\) satisfying \(I(t_{3})=\sigma\) and \(I(t)>\sigma\) for all \(t\in(t_{3}, t_{2})\). Then we can choose an integer \(l_{1} \geq0\) such that \(t_{2}\in[t_{3}+l_{1}\lambda_{0},t_{3}+(l_{1}+1)\lambda_{0})\) and integrate (2.20) from \(t_{3}\) to \(t_{2}\), and we have
It is a contradiction. Therefore, (2.21) holds.
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])
The system (1.2) is said to be globally attractive if any two solutions \((S_{1}(t),I_{1}(t),P_{1}(t))\) and \((S_{2}(t),I_{2}(t),P_{2}(t))\) of system (1.2) with initial conditions (1.3) satisfy
Theorem 2.4
Under assumptions (B1), (B2), (B3), if there exist constants \(\mu_{i}>0\) (\(i=1,2,3\)) satisfying \(\liminf_{t\rightarrow\infty} A_{i}(t)>0\), where
then system (1.2) is globally attractive.
Proof
Let \(x=S+I\), then model (1.2) can be rewritten as follows:
Suppose that \((x_{1}(t),I_{1}(t),P_{1}(t))\), \((x_{2}(t),I_{2}(t),P_{2}(t))\) are two arbitrary solutions of model (2.23). By (2.2), (2.3), we obtain
Define a Liapunov function
Then we have
Applying the conditions \(\liminf_{t\rightarrow \infty} A_{i}(t)>0 \) (\(i=1,2,3\)) and the definition of the inferior limit, we see that there are constants \(\bar{\alpha}>0\) and \(T^{\diamond}>0\) such that \(A_{i}(t)\geq\bar{\alpha}\) (\(i=1,2,3\)) for all \(t\geq T^{\diamond}\). Thus we obtain
for all \(t\geq T^{*}\). Integrating (2.25) from \(T^{*}\) to t, we obtain
therefore,
At the same time, by (2.23), (2.24), it can be seen that \(\frac{d}{dt}(x_{1}x_{2})\), \(\frac{d}{dt}(I_{1}I_{2})\), \(\frac {d}{dt}(P_{1}P_{2})\) are all bounded on \([0,\infty)\). By (2.26), we see that
The proof is completed. □
Remark 1
For model (1.1), we can also give the condition of the global attractivity for this SI model without predation as that for model (1.2) in Theorem 2.4, that is, if
then system (1.1) is globally attractive.
Some results for the periodic system
If model (1.2) is an ωperiodic system, then assumptions (B1), (B2), (B4) can degenerate into the following forms:

(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)
\(\overline{\beta}>0\).
Then we have some results for the periodic system as shown below.
Corollary 3.1
Under assumptions (A1), (A2), (B3), if
then the infective prey population I of model (1.2) is permanent.
Corollary 3.2
Under assumptions (A1), (A2), (B3), (A4), if
then the infective prey population I of model (1.2) goes to extinction.
Corollary 3.3
Under assumptions (B1), (B2), if \(d^{v}>0\), \(w_{1}^{v}>0\), \((\frac{c_{3}}{w_{2}})^{v}>0\), and there exists a constant \(\lambda >0\) satisfying
then the infective prey population I of model (1.2) is permanent.
Corollary 3.4
Under assumptions (B1), (B2), if \(d^{v}>0\), \(w_{1}^{v}>0\), \((\frac{c_{3}}{w_{2}})^{v}>0\), and there exist constants \(\lambda^{*} >0\), \(\lambda>0\) satisfying
then the infective prey population I of model (1.2) goes to extinction.
Remark 2
For model (1.1) without predation, assumptions (B1), (B2), (B4) are equivalent to the following forms:

(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)
\(\overline{\beta}>0\).
If assumptions (D1), (D2), (B3), (D4) hold, then from Corollaries 3.1 and 3.2, we can obtain the threshold value between extinction and permanence of the infective population in system (1.1), that is,

(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.
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].
First, for model (1.2), we choose the parameters \(\Lambda(t)=0.5+0.3\sin t\), \(d(t)=0.6+0.2\sin(2t)\), \(e(t)=0.3+0.2\sin t\), \(f(t)=0.05+0.04\sin t\), \(r(t)=0.5+0.4\sin t\), \(\alpha(t)=0.2+0.01\sin t\), \(\gamma (t)=0.05+0.01\sin t\), \(w_{1}(t)=8+0.5\cos t\), \(w_{2}(t)=0.2+0.08\sin t\), \(\sigma(t)=0.8+0.1\sin t\), \(c_{1}(t)=0.2+0.1\sin t\), \(c_{2}(t)=0.4+0.1\sin t\), \(c_{3}(t)=0.3+0.1\sin t\). Then assumptions (C1), (C2), and (B3) hold. Let \(\beta(t)=0.36+0.1\sin t\), by calculation we see that the upper threshold value \(R^{*}=0.9062<1\), which satisfies the conditions in Corollary 3.2. Thus, the infected prey population will go to extinction (see Figure 1). Then, let the infective rate increase to \(\beta(t)=0.9+0.1\sin t\), being similar to the above calculation, we can obtain the lower threshold value \(R_{*}=2.2271>1\) and see that model (1.2) is permanent from Figure 2, which verifies the conclusion of Corollary 3.1.
Second, let \(d(t)=0.5+0.1\sin t\), \(e(t)=1.2+0.2\sin t\), \(\gamma(t)=0.3+0.01\sin t\), \(\beta(t)=1.8+0.1\sin t\), and the other parameters are the same as in Figure 1. Considering system (1.2) with initial conditions \((0.5,1.7,0.6) \), \((0.01,0.07,0.09) \), \((0.2,1.1,0.03) \), \((0.03,0.06,0.02) \), \((0.3,0.3,0.3) \). From Figure 3, we can see that system (1.2) is globally attractive.
Third, we will study the role of predation on system dynamics through making a comparison between model (1.2) and (1.1).
Now, let \(\beta(t)=0.38+0.1\sin t\) and retain the other parameter values as in Figure 1, then we can see that the two upper threshold values are \(\widehat{R^{*}}=0.9987<1\), \(R^{*}=0.9528<1 \) for models (1.1) and (1.2), respectively, which shows that the infected prey I goes to extinction for both models (see Figure 4). Observing that \(\widehat{R^{*}}>R^{*}\), which means we have predation, the infected prey I in model (1.2) will be extinct more early and easily than in system (1.1). The results can also be observed from Figure 5, in which we choose \(\beta(t)=0.58+0.01\sin t\), \(c_{3}(t)=0.08+0.01\sin t\), \(e(t)=0.4+0.02\sin t\), and we can obtain the threshold values, \(\widehat{R^{*}}=1.1019\), \(R^{*}=0.9567 \), for models (1.1) and (2.2). The figure shows that model (1.2) is disease free, while the infected prey population for model (1.1) without predation is permanent. Then we conclude that the predator can be used as a biocontroller to keep the model disease free.
Next, let the infection rate increase to \(\beta(t)=0.8+0.1\sin t\), we can easily get \(\widehat{R_{*}}=2.0255>1\), \(R_{*}=1.9611>1\) and \(\widehat{R_{*}}>R_{*}\) for models (1.1) and (1.2), respectively. Then we can observe that all the species of system (1.1) and (1.2) enter into a steady state from Figure 6. Therefore, we could conclude that predation is benefit for controlling disease and enhancing permanence in a predatorprey model.
Fourth, some discussions are given for the intermediate case where \(R^{*}>1\) while \(R_{*}\leq1\). Choose the infection rate \(\beta(t)=0.438+0.1\sin t\) and retain the other parameter values as in Figure 1, then we can see that the two threshold values are \(R^{*}=1.0881>1\) and \(R_{*}=0.9982<1 \) for model (1.2). From Figure 7, it can be shown that the infected prey I goes to extinction. Changing the infection rate from \(\beta(t)=0.53+0.1\sin t\) to \(\beta(t)=0.73+0.1\sin t\), then we have the upper threshold values \(R^{*}=1.3027\) and \(R^{*}=1.7692\), respectively, which are also greater than 1, however, by Figure 8(a)(b), it can be seen that the infected prey population is permanent. From Figure 7, it could be concluded that if the lower threshold value \(R^{*}\leq1\), the infected prey could go to extinction. In addition, comparing Figure 7(a) and Figure 8, it could be shown that the condition we obtained for the extinction of the infected prey is only a sufficient condition. What is the sufficient necessary condition? This will be left as our future consideration.
Last but not least, we turn to the role of the functional response in controlling disease. In [15], we considered a predatorprey model with a linear predation rate for migratory birds, that is,
by theoretical analysis, we showed that if \(R^{\prime *}=\frac{\overline{\beta\widetilde{S_{0}}/(1+\alpha \widetilde{S_{0}})}}{(\overline{e}+\overline{f}+\overline{k_{2} \tilde{y}_{0}})} \leq1 \), then the infective prey population of system (4.1) goes to extinction. However, in this paper, we assume that the predator eat both the susceptible and the infected prey population with modified LeslieGower schemes and we obtain the upper threshold value \(R^{*}\) to determine the extinction of the infection. Now we give a numerical simulation to study the effects of different predation rates in controlling the disease. Let \(\Lambda(t)=0.2+0.1\sin t\), \(d(t)=0.5+0.2\sin(2t)\), \(e(t)=0.6+0.2\sin(2t)\), \(f(t)=0.05+0.045\sin(2t)\), \(r(t)=0.5+0.4\sin t\), \(\beta(t)=0.7+0.1\sin t\), \(K(t)=0.1+0.08\sin t\), \(w_{1}(t)=2+0.5\cos t\), \(w_{2}(t)=0.1+0.08\sin t\), \(c_{1}(t)=k_{1}(t)=0.2+0.08\sin t\), \(c_{2}(t)=k_{2}(t)=1+0.5\cos t\), \(c_{3}(t)=r(t)=0.5+0.4\sin t\), \(\gamma(t)=0.2+0.1\sin t\), \(k_{1}'(t)=1.1+\sin t\), \(k_{2}'(t)=0.2+0.08\cos t\), \(\sigma (t)=0.8+0.1 \sin t\), and \(\alpha(t)=0.2+0.1\sin t\) for system (4.1) while \(\alpha(t)=0\) for system (1.2). Then we can obtain the upper threshold values \(R^{*}=0.4226<1\) for model (1.2) and \(R^{\prime *}=0.8836<1\) for model (4.1), from which we see that the infected prey I goes to extinction for both models, and Figure 9 confirms it. Moreover, obviously, \(R^{*}< R^{\prime *}\) and we can conclude that the modified LeslieGower functional predation rate may be a good choice that can be used to control the disease more easily and effectively.
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.
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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).
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Fan, K., Zhang, Y. & Gao, S. On a new ecoepidemiological model for migratory birds with modified LeslieGower functional schemes. Adv Differ Equ 2016, 97 (2016). https://doi.org/10.1186/s1366201608253
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Keywords
 migratory birds
 LeslieGower functional response
 saturated incidence and recovery rate