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Convergences of a stagestructured predatorprey model with modified LeslieGower and Hollingtype II schemes
Advances in Difference Equations volume 2016, Article number: 181 (2016)
Abstract
A stagestructured predatorprey model (stage structure for both predator and prey) with modified LeslieGower and HollingII schemes is studied in this paper. Using the iterative technique method and the fluctuation lemma, sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibrium are obtained. Our results indicate that for a stagestructured predatorprey community, both the stage structure and the death rate of the mature species are the important factors that lead to the permanence or extinction of the system.
Introduction
Since the pioneer work of Aiello and Freedman [1], the stagestructured population models have been investigated extensively and many excellent results have been obtained (see [1–35]). Recently, Huo et al. [5] considered a stagestructured predatorprey model with modified LeslieGower and Hollingtype II schemes as follows:
where \(x_{1}\), \(x_{2}\), and y represent the population densities of immature prey, mature prey and predator at time t, respectively; \(r_{1}\) is the birth rate of immature prey \(x_{1}\); \(d_{11}\) denotes the death rate of the immature prey \(x_{1}\); \(r_{2}\) is the intrinsic growth rate of predator y; b represents the strength of intraspecific competition in the mature prey; \(a_{1}\) represents the maximum value that mature \(x_{2}\) can be captured by predator y, and the meaning of \(a_{2}\) is similar to \(a_{1}\); \(k_{1}\) and \(k_{2}\) measure the protection degree that the environment could afford for prey \(x_{2}\) and predator y, respectively; \(\tau_{1}\) is the time to maturity for prey; \(r_{1}e^{d_{11}\tau_{1}}x_{2}(t\tau_{1})\) represents the prey who were born at time \(t\tau_{1}\) and survive and become mature at time t.
In [5], the authors analyzed the dynamics of system (1.1), specially, by using the iterative technique, the authors obtained a set of sufficient conditions which guarantee the existence of a unique globally attractive positive equilibrium. Li et al. [6] found that some of the conditions in [5] are redundant, and they obtained the following result.
Theorem A
Suppose that
holds, then the system (1.1) has a unique globally attractive positive equilibria E.
As we can see, [5] and [6] only considered the stage structure of immature and mature of the prey species, yet they ignored that of the predator ones. Already, several scholars had proposed and investigated the dynamic behaviors of the predatorprey system with stage structure for predator species [8–11, 17, 24–26]. Indeed, Wang and Chen [17] considered the following predatorprey system with stage structure for the predator population:
In [17], the authors studied the asymptotic behavior of the above system. When a time delay due to gestation of the predators and a time delay from a crowding effect of the prey are incorporated, they establish conditions for the permanence of the populations and sufficient conditions under which a positive equilibrium of the above system is globally stable; Zhang and Luo [8] argued that above system is not a realistic model because it is an autonomous system, and they incorporated a type IV functional response into above system; by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for the system is established. Recently, Chen et al. [10, 11] studied the persistence and extinction property of the following stagestructured predatorprey system (stage structure for both predator and prey, respectively):
where \(x_{1}(t)\) and \(x_{2}(t)\) denote the densities of the immature and mature prey species at time t, respectively; \(y_{1}(t)\) and \(y_{2}(t)\) represent the immature and mature population densities of predator species at time t, respectively; \(r_{i}(t)\), \(b_{i}(t)\), \(c_{i}(t)\) (\(i=1,2\)) are all continuous functions bounded above and below by positive constants for all \(t \geq0\). \(d_{ij}\), \(\tau_{i}\), \(i,j=1,2\) are all positive constants. There are many interesting properties of this system, for example, due to the influence of the stage structure, the extinction of the predator species could not directly imply the permanence of the prey species. The extinction of the prey species could not lead to the extinction of the predator species. Under certain assumptions, the system would be broken, which means that both predators and prey species would be driven to extinction. However, all the works of [9–11] did not take the functional response of the predator species into consideration.
Now, stimulated by the work of [5, 6, 9–11], we consider the following stagestructured predatorprey model (stage structure for both predator and prey, respectively) with modified LeslieGower and Hollingtype II schemes:
where \(d_{12}\) and \(d_{21}\) represent the death rate of mature prey \(x_{2}\) and mature predator \(y_{2}\), respectively; \(\tau_{1}\) is the time length of the prey species from immature ones to mature ones, \(\tau_{2}\) is the time length of the predator from immature ones to mature ones. Other parameters have the same biological meaning as that of system (1.1). All parameters are positive constants in system (1.2).
The initial conditions for system (1.2) take the form of
where \(\tau=\max{\{\tau_{1},\tau_{2}\}}\). For continuity of the initial conditions, we assume that
Integrating both sides of the first and third equation of system (1.2) (see [12]) over the interval \((0,t)\) leads to
This suggests that the dynamics of model (1.1) is completely determined by its second and fourth equations. Therefore, in the rest of this paper, we investigate the asymptotic behavior for the subsystem of system (1.1) as follows:
The organization of this paper is as follows: The main results are stated and proved in Sections 2 and 3, respectively. In Section 4, several examples together with their numerical simulations are presented to illustrate the feasibility of our main results. We end this paper by a brief discussion. For more work on the LeslieGower predatorprey system, one may refer to [36–39] and the references cited therein.
Main results
For convenience, we denote
Let \(x'_{2}(t)=0\), \(y'_{2}(t)=0\) in system (1.6), we can get four equilibria as follows:
where E is an interior equilibrium point in system (1.6). The components of E are given by
where \(x^{*}_{2}\) is a positive solution of the second order equation as follows:
where \(C=a_{1}k_{2}\lambda_{2}a_{2}k_{1}\lambda_{1}\), we can see that there exists a unique \(x_{2}^{*}>0\) if \(C<0\), i.e.,
\(E_{1}\), \(E_{2}\) are two of the boundary equilibria of the system (1.6) if \(\lambda_{1}>0\), \(\lambda_{2}>0 \).
Consequently, we have the following theorem.
Theorem 2.1
Assume that inequality (2.1) holds, then system (1.6) admits a unique positive equilibrium point E.
Theorem 2.2
Suppose that
hold, then the unique positive equilibrium E is globally attractive.
Remark 2.1
If \(d_{12}=d_{21}=0\), \(\tau_{2}\neq0\), that is, we only consider the stage structure of the predator species and ignore the death rate of the mature predator and prey species, in this case, (\(\mathrm{H}_{1}\)) holds naturally, and \(\lambda_{3}\) in condition (\(\mathrm{H}_{2}\)) would reduce to
Noting that \(r_{2}e^{d_{22}\tau_{2}}< r_{2}\), (\(\mathrm{H}_{2}'\)) is weaker than (H) in Theorem A, this means that the stage structure of predator species has benefit for the coexistence of the system.
Remark 2.2
Suppose that \(\tau_{2}= 0\), i.e., we did not consider the stage structure of the predator species, then \(\lambda_{2}\) and \(\lambda_{3}\) in Theorem 2.2 become \(\lambda_{2}=r_{2}d_{21}>0\), \(\lambda_{3}=\lambda_{0}  (a_{2}k_{1}ba_{1}r_{2}+a_{1}d_{21})d_{12}+(a_{1}k_{2}b+a_{1}r_{1}e^{d_{11}\tau _{1}})d_{21}>0\), respectively.

(i)
If \(d_{12}= 0\), \(d_{21}\neq0\), in this case, \(\lambda _{3}>\lambda_{0}\), that is, introducting the mortality item of the predator species improving the coexistence rate of the two species.

(ii)
If \(d_{12}\neq0\), \(d_{21}= 0\), in this case, \(\lambda _{0}>\lambda_{3}\), that is, introducting the mortality item of the prey species decreasing the chance of coexistence of both species.
Theorem 2.3
Suppose that
holds, then both of the predator and prey species will be driven to extinction, that is, \(E_{0}\) is globally attractive.
Theorem 2.4
Suppose that
holds, then \(E_{1}\) is globally attractive.
Remark 2.3
From [5], we know that \(E_{0}(0,0,0)\) of the system (1.1) is unstable, which implies the extinction of both predator and prey species is impossible. However, if the death rates of the mature prey and predator species are large enough, (\(\mathrm{H}_{3}\)) in Theorem 2.3 would hold, and consequently both the prey and the predator species will be driven to extinction. By constructing a suitable Lyapunov function, Korobeinikov [27] showed that the unique positive equilibrium of the traditional LeslieGower predatorprey model is globally attractive, which means that it is impossible for the predator species to become extinct. However, Theorem 2.4 shows that if the death rate of the mature predator species is large enough, (\(\mathrm{H}_{4}\)) would hold and the predator species will be driven to extinction. Theorems 2.3 and 2.4 show that the death rates of the mature predator and prey species are two of the essential factors to determine the persistent property of the system.
Theorem 2.5
Suppose that
holds, then \(E_{2}\) is globally attractive.
Corollary 2.1
If the parameters of system (1.2) satisfy the condition (2.1), then system (1.2) has a unique positive equilibrium point \(E'(x^{*}_{1},x^{*}_{2},y^{*}_{1},y^{*}_{2})\), where \(x^{*}_{1}= \frac{r_{1}x^{*}_{2}(1e^{d_{11}\tau_{1}})}{d_{11}}\), \(y^{*}_{1}=\frac{r_{2}y^{*}_{2}(1e^{d_{22}\tau_{2}})}{d_{22}} \).
Corollary 2.2
If the parameters of system (1.2) satisfy the conditions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)), then \(E'\) is globally attractive.
Corollary 2.3
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{3}\)), then \(E'_{0}=(0,0,0,0)\) is globally attractive.
Corollary 2.4
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{4}\)), then \(E'_{1}=(x_{1*},x_{2*},0,0)\) is globally attractive, where \(x_{1*}=\frac{r_{1}x_{2*}(1e^{d_{11}\tau_{1}})}{d_{11}}\).
Corollary 2.5
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{5}\)), then \(E'_{2}=(0,0,y_{1*},y_{2*})\) is globally attractive, where \(y_{1*}=\frac{r_{2}y_{2*}(1e^{d_{22}\tau_{2}})}{d_{22}}\).
Proof of the main results
Now let us state several lemmas which will be useful in proving the main results.
Lemma 3.1
Assume that \(x_{2}(\theta)\geq0\), \(y_{2}(\theta)\geq0\) are continuous on \(\theta\in[\tau,0]\), and \(x_{2}(0)>0\), \(y_{2}(0)>0\). Let \((x_{2}(t),y_{2}(t))^{T}\) be a any solution of system (1.6), then \(x_{2}(t)>0\), \(y_{2}(t)>0\) for all \(t>0\).
The proof of Lemma 3.1 is similar to the proof of Theorem 1 in [1], so we omit its proof.
Lemma 3.2
[2]
Consider the following equation:
and assume that \(b, a_{2}>0\), \(a_{1}\geq0\), and \(\delta\geq0\) is a constant. Then

(i)
if \(b\geq a_{1}\), then \(\lim_{t\rightarrow+\infty }x(t)= \frac{ba_{1}}{a_{2}}\);

(ii)
if \(b\leq a_{1}\), then \(\lim_{t\rightarrow+\infty}x(t)=0\).
Lemma 3.3
(Fluctuation lemma [23])
Let \(x(t)\) be a bounded differentiable function on \((\alpha,\infty)\), Then there exist sequences \(\gamma_{n}\rightarrow\infty\), \(\sigma _{n}\rightarrow\infty\) such that

(i)
\(x'(\gamma_{n})\rightarrow0 \) and \(x(\gamma_{n})\rightarrow\limsup_{t\rightarrow+\infty }x(t)=\overline{x}\) as \(n\rightarrow\infty\),

(ii)
\(x'(\sigma_{n})\rightarrow0 \) and \(x(\sigma_{n})\rightarrow\liminf_{t\rightarrow+\infty }x(t)=\underline{x}\) as \(n\rightarrow\infty\).
Lemma 3.4
Assume that \(x_{2}(\theta), y_{2}(\theta )\geq0\) are continuous on \(\theta\in[\tau,0]\), and \(x_{2}(0)>0\), \(y_{2}(0)>0\). Let \((x_{2}(t),y_{2}(t))^{T}\) be a any solution of system (1.6). If \(\lambda_{2}>0\), then
Proof
From the second equation of system (1.6), we have
Since \(\lambda_{2}>0\), and by applying Lemma 3.2(i), and the standard comparison theorem, we have
This completes the proof of Lemma 3.4. □
Now we start to prove the above results.
Proof of Theorem 2.2
From the first equation of system (1.6), we have
According to condition (\(\mathrm{H}_{1}\)), we know \(r_{1}e^{d_{11}\tau_{1}}d_{12}>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
So, for any small constant \(\varepsilon>0\), there exists a \(T_{1}>0\) such that
For \(t>T_{1}+\tau_{2}\), substituting (3.1) into the second equation of system (1.6), we have
According to condition (\(\mathrm{H}_{1}\)), we have \(r_{2}e^{d_{22}\tau_{2}}d_{21}>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then, for the above ε, there exists a \(T_{2}>T_{1}+\tau_{2}\), such that
For \(t>T_{2}+\tau_{1}\), substituting (3.2) into the first equation of system (1.6), we have
Let
Then substituting (3.1) and (3.2) into Δ, we have
Then, for small enough \(\varepsilon>0\) and condition (\(\mathrm{H}_{2}\)), we have
By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then, for the above \(\varepsilon>0\), there exists a \(T_{3}>T_{2}+\tau_{1}\), such that
For \(t>T_{3}+\tau_{2}\), substituting (3.4) into the second equation of system (1.6), we have
By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then for the above \(\varepsilon>0\), there exists a \(T_{4}>T_{3}+\tau_{2}\), such that
According to (3.1), (3.2), (3.4), and (3.5), we obtain
Then for \(t>T_{4}+\tau_{1}\), substituting (3.1) and (3.5) into the first equation of system (1.6), we have
According to the inequalities (3.3) and (3.6), we have \(r_{1}e^{d_{11}\tau_{1}} (d_{12}+\frac {a_{1}m^{(1)}_{2}}{k_{1}+M^{(1)}_{1}} )>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then, for the above \(\varepsilon>0\), there exists a \(T_{5}>T_{4}+\tau_{1}\), such that
From inequalities (3.1) and (3.7), we obtain
For \(t>T_{5}+\tau_{2}\), substituting (3.7) into the second equation of system (1.6), we have
By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then for above \(\varepsilon>0\), there exists a \(T_{6}>T_{5}+\tau_{2}\), such that
From inequalities (3.2), (3.8), and (3.9), we have
For \(t>T_{6}+\tau_{1} \), substituting inequalities (3.4) and (3.9) into the first equation of system (1.6), we have
According to inequalities (3.3) and (3.10), we can obtain \(r_{1}e^{d_{11}\tau_{1}}d_{12}\frac{a_{1}M^{(2)}_{2}}{k_{1}+m^{(1)}_{1}}>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then, for the above \(\varepsilon>0\), there exists a \(T_{7}>T_{6}+\tau_{1}\), such that
According to the inequalities (3.4), (3.10), and (3.11), we can obtain
Substituting inequality (3.11) into the second equation of system (1.6), we have
By applying Lemma 3.2(i), and the standard comparison theorem, we have
Then, for the above \(\varepsilon>0\), there exists a \(T_{8}>T_{7}+\tau_{2}\), such that
According to the inequalities (3.5), (3.12), and (3.13), we can obtain
For \(t>T_{8}\), according to (3.8), (3.10), (3.12), and (3.14), we have
Repeating the above process, we get four sequences
For \(i=1,2\), we claim that \(M^{(n)}_{i}\) are monotonic decreasing sequences, and \(m^{(n)}_{i}\) are monotone increasing sequences. In the following we will prove this claim by induction. First of all, according to inequalities (3.15), we have
Second, we suppose that our claim is true for n, that is,
Noting that
According to inequalities (3.16), (3.17), and (3.18), one could easily see that
Then for \(t>T_{4n}\), we have
Therefore the limits of \(M^{(n)}_{i}\), \(m^{(n)}_{i}\) (\(i=1,2\), \(n=1,2,\ldots\)) exist. Denote that
Consequently, \(\overline{x}_{2} \geq\underline{x}_{2}\), \(\overline{y}_{2} \geq\underline{y}_{2}\). In order to complete the proof, we just need to show that \(\overline {x}_{2}= \underline{x}_{2}\), \(\overline{y}_{2}=\underline{y}_{2}\). Letting \(n\rightarrow+\infty\) in (3.16), we have
It follows from the above four equations that
Subtracting the first equation of (3.19) from the second equation, we get
Suppose that \(\overline{x}_{2}\neq\underline{x}_{2}\), it follows from the above equation that
Substituting (3.20) into (3.19), we find \(\overline {x}_{2} \) and \(\underline{x}_{2}\) both satisfy the following equation:
Simplifying equality (3.21), we get
where
According to condition (\(\mathrm{H}_{2}\)), we can immediately obtain
It implies that
Therefore, we have \(D<0\), that is, equation (3.22) has only one positive root. Then \(\overline{x}_{2}=\underline{x}_{2}\), and consequently, \(\overline{y}_{2}=\underline{y}_{2}\). Obviously, conditions (\(\mathrm{H}_{1}\)), (\(\mathrm{H}_{2}\)) imply inequality (2.1), so system (1.6) has a unique positive equilibrium \(E(x^{*}_{2},y^{*}_{2})\). That is,
This completes the proof of Theorem 2.2. □
Proof of Theorem 2.3
It follows from the first equation of system (1.6) that
According to first inequality of condition (\(\mathrm{H}_{3}\)), we have \(r_{1}e^{d_{11}\tau_{1}}d_{12}<0\). By applying Lemma 3.2(ii) and the standard comparison theorem, we have \(\limsup_{t\rightarrow+\infty}x_{2}(t)\leq0\). That is,
Then, for any \(\varepsilon>0\), there exists a \(T>0\) such that
Therefore, it follows from the second equation of system (1.6) that
Similar to the above analysis, we also have
Therefore, \(E_{0}=(0,0)\) is globally attractive. This completes the proof of Theorem 2.3. □
Proof of Theorem 2.4
According to the first inequality of condition (\(\mathrm{H}_{4}\)), we have \(r_{1}e^{d_{11}\tau_{1}}d_{12}>0\). Therefore from the proof of Theorem 2.2, we know that
And for any small positive constant \(\varepsilon>0\), there exists a \(T_{1}>0\) such that
According the second inequality of condition (\(\mathrm{H}_{4}\)), we have \(r_{2}e^{d_{22}\tau_{2}}d_{21}<0\). By applying Lemma 3.2(ii) and the standard comparison theorem, we have \(\limsup_{t\rightarrow+\infty}y_{2}(t)\leq0\). That is,
Then, for any small \(\varepsilon>0\), there exists \(T_{2}>T_{1}+\tau_{2}\), such that
Substituting inequality (3.24) into the first equation of system (1.6), we have
Since \(r_{1}e^{d_{11}\tau_{1}}d_{12}>0\), we can choose sufficiently small \(\varepsilon>0\) such that \(r_{1}e^{d_{11}\tau _{1}}d_{12} \frac{a_{1}\varepsilon}{k_{1}}>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
For the above formula, letting \(\varepsilon\rightarrow0\), we have
From inequalities (3.23) and (3.25), we get
Then we have
Therefore, \(E_{1}(x_{2*},0)\) is globally attractive. This completes the proof of Theorem 2.4. □
Proof of Theorem 2.5
According to the fluctuation lemma, there exist two sequences \(\gamma_{n}\rightarrow\infty\), \(\sigma_{n}\rightarrow\infty\) such that \(x'_{2}(\gamma_{n})\rightarrow0\), \(x_{2}(\gamma_{n})\rightarrow\limsup_{t\rightarrow+\infty}x(t)=\overline{x}_{2}\), and \(y'_{2}(\sigma_{n})\rightarrow0 \), \(y_{2}(\sigma_{n})\rightarrow\liminf_{t\rightarrow+\infty}y_{2}(t)=\underline{y}_{2}\) as \(n\rightarrow\infty\). Since from Lemma 3.1, we know \(\overline{x}_{2}\geq0\). In order to prove \(\lim_{t\rightarrow\infty}x_{2}(t)=0\), we only need to prove \(\overline{x}_{2}=0\), so for getting a contradiction, we suppose that \(\overline{x}_{2}>0\). Since for \(\lambda_{2}>0\), and according to Lemma 3.4, we know \(\underline{y}_{2}>0\), it follows from the first equation of the system (1.6) that
Letting \(n\rightarrow\infty\) in the above inequality, we obtain
that is,
From the second equation of system (1.6), by a similar argument, we have
It follows from inequalities (3.26) and (3.27) that
Simplifying the above inequality, we have
According to the second inequality of condition (\(\mathrm{H}_{5}\)), we know \(a_{1}k_{2}\lambda_{2}a_{2}k_{1}\lambda_{1}>0\), \(a_{2}k_{1}ba_{2}\lambda_{1}>0\), then only \(\overline{x}_{2}<0 \) can ensure (3.28) holds. And \(\overline{x}_{2}<0\) contradicts the hypothesis \(\overline{x}_{2}>0\), then we get
Then, for any small enough \(\varepsilon>0\), there exists a \(T>0\), such that
Substituting inequality (3.29) into the second equation of system (1.6), we have
According to second inequality of condition (\(\mathrm{H}_{5}\)), we know \(r_{2}e^{d_{22}\tau_{2}}d_{21}>0\). By applying Lemma 3.2(i), and the standard comparison theorem, we have
Letting \(\varepsilon\rightarrow0\) in the above inequality, we have
According to Lemma 3.4 and (3.30), we obtain
Then we have
This completes the proof of Theorem 2.5. □
Numerical simulations
The following examples show the feasibility of our main results.
Example 4.1
where \(r_{1}=2\); \(r_{2}=1.2\); \(d_{11}=0.4\); \(d_{22}=0.5\); \(\tau_{1}=1\); \(\tau_{2}=1\); \(d_{12}=0.4\); \(d_{21}=0.6\); \(b=0.8\); \(a_{1}=1\); \(a_{2}=2\); \(k_{1}=1\); \(k_{2}=2\). One could easily verify that \(\lambda_{1}\approx r_{1}e^{d_{11}\tau_{1}}d_{12}=0.9406>0\), \(\lambda _{2}\approx r_{2}e^{d_{22}\tau_{2}}d_{21}=0.1278>0\), \(\lambda_{3}\approx b(a_{2}k_{1}\lambda_{1}a_{1}k_{2}\lambda_{2}) a_{1}\lambda_{1}\lambda_{2} =1.1802>0\), which shows conditions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)) hold. According to Theorem 2.2, system (1.6) has a unique and globally attractive positive equilibrium \(E(1.0571,0.1954)\). Figure 1 indicates the dynamical behavior of system (4.1).
Example 4.2
where \(r_{1}=1\); \(r_{2}=1.2\); \(d_{11}=0.5\); \(d_{22}=0.7\); \(\tau_{1}=1\); \(\tau_{2}=1\); \(d_{12}=0.7\); \(d_{21}=0.8\); \(b=0.8\); \(a_{1}=1\); \(a_{2}=2\); \(k_{1}=1\); \(k_{2}=2\). By simple computation, one could see that \(\lambda_{1}\approx0.0935<0\), \(\lambda _{2}\approx0.2041<0\), which shows condition (\(\mathrm{H}_{3}\)) holds. It follows from Theorem 2.3 that the solution \(E_{0}(0,0)\) of system (4.2) is globally attractive. Figure 2 shows the feasibility of this case.
Example 4.3
where \(r_{1}=2.2\); \(r_{2}=1.5\); \(d_{11}=0.5\); \(d_{22}=0.4\); \(\tau_{1}=2\); \(\tau_{2}=2\); \(d_{12}=0.7\); \(d_{21}=0.8\); \(b=0.8\); \(a_{1}=1\); \(a_{2}=2\); \(k_{1}=1\); \(k_{2}=2\). By computation, we have \(\lambda_{1}\approx0.1093 >0\), \(\lambda_{2}\approx 0.1260<0\), which shows condition (\(\mathrm{H}_{4}\)) holds. It follows from Theorem 2.4 that the boundary equilibrium \(E_{1}(0.1367,0)\) of system (4.3) is globally attractive. Figure 3 supports this assertion.
Example 4.4
where \(r_{1}=1.5\); \(r_{2}=1.5\); \(d_{11}=0.4\); \(d_{22}=0.5\); \(\tau_{1}=1\); \(\tau_{2}=1\); \(d_{12}=0.8\); \(d_{21}=0.6\); \(b=0.5\); \(a_{1}=1\); \(a_{2}=1.5\); \(k_{1}=0.6\); \(k_{2}=2\). By computation, we have \(\lambda_{1}\approx0.2055\), \(\lambda_{2}\approx 0.3098>0\), \(\lambda_{1}\frac{a_{1}k_{2}\lambda_{2}}{a_{2}k_{1}}\approx 0.9945<0\), \(\lambda_{1}k_{1}b\approx0.0945<0\), which shows condition (\(\mathrm{H}_{5}\)) holds. It follows from Theorem 2.5 that the solution \(E_{2}(0, 0.4131)\) of system (4.4) is globally attractive. Figure 4 shows the feasibility of this case.
Example 4.5
where \(r_{1}=2\); \(r_{2}=1.5\); \(d_{11}=0.5\); \(d_{22}=0.7\); \(\tau_{1}=1\); \(\tau_{2}=1\); \(d_{12}=0.7\); \(d_{21}=0.6\); \(b=0.8\); \(a_{1}=1\); \(a_{2}=2\); \(k_{1}=1\); \(k_{2}=2\). And so \(a_{2}k_{1}\lambda_{1}\approx1.0261>0.2898\approx a_{1}k_{2}\lambda_{2}>0\), which shows condition (2.1) holds. However, \(\lambda_{3}\approx0.3145<0\), and so, condition (\(\mathrm{H}_{2}\)) does not hold. A numeric simulation (Figure 5) shows that the system still admits a unique globally attractive positive equilibrium \(E(0.4900,0.1804)\).
Conclusion
Huo et al. [5] and Li et al. [6] studied the stability property of the positive equilibrium of a stagestructured predatorprey model with modified LeslieGower and Hollingtype II schemes. In those two papers, the authors only consider the stage structure of prey species and ignore that of predator species. Stimulated by [9–11], we consider a model with stage structure for both predator and prey species. By applying an iterative technique and the fluctuation lemma, sufficient conditions which guarantee the global attractivity of all the nonnegative equilibria are obtained. Our study indicates that both the stage structure of the species and the death rate of the mature predator and prey species are the important factors on the dynamic behaviors of the system. If the death rates of the mature prey and predator species are too large or the degree of the stage structure of the species is large enough, then at least one of the species will be driven to extinction. We would like to mention here that Example 4.5 shows that our result in Theorem 2.2 has room for improvement. We conjecture that condition (2.1) is enough to ensure the global attractivity of the positive equilibrium. We leave this for future work.
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Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. The research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019, 2015J05006) and the Scientific Research Foundation of Fuzhou University (XRC1438).
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Lin, Y., Xie, X., Chen, F. et al. Convergences of a stagestructured predatorprey model with modified LeslieGower and Hollingtype II schemes. Adv Differ Equ 2016, 181 (2016). https://doi.org/10.1186/s1366201608872
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DOI: https://doi.org/10.1186/s1366201608872
MSC
 34D23
 92B05
 34D40
Keywords
 global stability
 extinction
 stagestructure
 LeslieGower
 Hollingtype II
 predatorprey