Abstract
In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.
1 Introduction
Predator-prey model is one of the basic models between different species in real world. As we all know, there are always many species going through two stages, immature and mature, which reflect the different characteristics of species at each stage. Therefore, to exhibit the real world phenomenon, stage structure population models are more reasonable than other models. In recent years, numerous papers have considered the predator-prey system with stage structure (see [1, 2, 3, 4, 5]).
On the other hand, in general, the consumption of prey by predator throughout its past history governs the present birth rate of the predator, in other words, time delay due to gestation is a common example. Obviously, delay differential equations exhibit much more complicated dynamics than ordinary differential equations (see [6, 7, 8, 9, 10, 11, 12, 13, 14]). For example, Wang and Chen [15] considered the following predator prey system with stage structure for the predator population:
The authors studied the asymptotic behavior of system (1.1). When a time delay due to gestation of the predators and a time delay from a crowding effect of the prey are incorporated, we establish conditions for the permanence of the populations and sufficient conditions for the existence of globally stable positive equilibrium of system (1.1).
Beddington [16] and DeAngelis et al. [17] established a famous B-D functional response that is a predator dependent functional response. In many cases, predators need to search for food and share or compete for food. Therefore, the stage-structured predator-prey models incorporating Beddington-DeAngelis functional response better reflect the ecology. Chen et al. [18] discussed the stability of the boundary solution of a nonautonomous predator-prey model with the Beddington-DeAngelis functional response, which reflects the dynamics of interacting predators and prey in a fluctuating environment. Xia et al. [19] considered stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting. Chen et al. [20] discussed the extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances. Khajanchi [21] investigated the dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model:
By analyzing the above system, conditions for positivity, boundedness, uniform persistence, existence of positive equilibria with their local stability have been established. Also, the author showed the existence of Hopf bifurcation when the conversion parameter k1 passes the critical value. Finally, the conditions for the occurrence of global stability for the unique interior equilibrium point were derived.
In the real world, refuge is a strategy to reduce the risk of predation. It is clear that the existence of refuge can have a significant impact on the coexistence of predator species and prey species. In recent years, many papers [22, 23, 24, 25, 26, 27] have proposed and analyzed predator-prey models incorporating prey refuges. Recently, Wei and Fu [28] discussed the Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refuge
By using the characteristic equations, the local stability of each feasible equilibrium of model (1.3) was discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium was established.
Motivated by the works [21] and [28], a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge is investigated in this paper. The proposed model is as follows:
where x(t), y1(t) and y2(t) denote the densities of prey species, immature predator species and mature predator species at time t, respectively; r is the intrinsic growth rate of prey species; a is the intraspecific competition rate of prey species; d and e are the death rates of the immature and mature predator species respectively; μ(1 – m) is the capturing rate of the mature predator; η is the conversion rate of nutrients into the production of predator species; τ denotes the time delay due to the gestation of the mature predator; m ∈ [0, 1) is refuge rate to prey; the predator species consumes the prey species with Beddington-DeAngelis functional response incorporating prey refuge
The initial conditions for system (1.4) take the form
where (ϕ(θ), ψ1(θ), ψ2(θ)) ∈ C([–τ, 0],
The rest of this paper is organized as follows. The boundedness and local stability of the equilibrium and the existence of Hopf bifurcation at positive equilibrium of system (1.4) are derived in the next section. In Section 3, we study the permanence of system (1.4). In Section 4, the global stability of system (1.4) are investigated. In Section 5, the influence of refuge rate on the densities to predator species and prey species is discussed. We end this paper with some examples and a briefly discussion.
2 Boundedness, Local stability and Hopf bifurcation
In this section, we study the boundedness and local stability of the equilibrium as well as the existence of Hopf bifurcation at positive equilibrium of system (1.4). It is obvious that solutions of model (1.4) with initial conditions (1.5) are positive for all t ≥ 0. The result is a direct consequence of Nagumo’s theorem [29].
2.1 Boundedness
Theorem 2.1
Every solution of system (1.4) with initial conditions (1.5) is bounded for all t ≥ 0 and all of these solutions are ultimately bounded.
Proof
Let V(t) = ηx(t – τ) + y1(t) + y2(t), and calculating the derivative of V(t) with respect to t along the positive solution of system (1.4), we have
For a small positive constant s ≤ min{d, e},
Hence there exists a positive constant
that is
Thus V(t) is ultimately bounded, that is, each solution z(t) = (x(t), y1(t), y2(t)) of system (1.4) is ultimately bounded. The proof is complete.
2.2 Equilibria
Obviously, system (1.4) always has a trivial equilibrium E0(0, 0, 0) and a predator-extinction equilibrium E1(r/a, 0, 0). Further, if the following holds:
then model (1.4) has a unique coexistence equilibrium
with
Let E͠ = (x̃, ỹ1, ỹ2) be any arbitrary equilibrium, then the variational matrix of system (1.4) at E͠ is given by
and the characteristic equation becomes
2.3 E0 = (0, 0, 0)
First, we analyze the stability of equilibrium E0.
Theorem 2.2
The trivial equilibrium E0(0, 0, 0) of system (1.4) is unstable.
2.4 E1 = (r/a, 0, 0)
After that, we consider the stability of equilibrium E1.
Theorem 2.3
If the following holds:
then the predator-extinction equilibrium E1(r/a, 0, 0) of system (1.4) is locally asymptotically stable; if (H1) holds, then E1 is unstable.
Proof
The characteristic equation (2.6) at predator-extinction equilibrium E1 becomes
Clearly, the equation λ + r = 0 has one negative real root, which implies that all other roots of Eq. (2.8) are determined by
where h1 = e + β + d > 0, h2 = e(β + d), h3 =
When τ = 0, Eq. (2.9) turns to
According to (H2), we have h2 + h3 > 0. By the Routh-Hurwitz criterion, the boundary equilibrium E1 is locally asymptotically stable. If (H1) holds, then Eq. (2.10) has at least a positive real root, thus the predator-extinction equilibrium E1 is unstable.
For τ > 0, we investigate the existence of purely imaginary roots of (2.9). If iω1(ω1 > 0) is a solution of (2.9) if and only if ω1 satisfies
Separating the real and imaginary parts, we obtain
which implies
Note that
and h2 + h3 > 0, then
2.5
E ∗ = ( x ∗ , y 1 ∗ , y 2 ∗ )
Further, we analyze the stability of equilibrium E*.
For the positive equilibrium
where
with
When τ = 0, Eq. (2.13) turns to
From (H1), we derive that
Denote
We see that if r < min{r2, r4} and (H1) hold, then p2 > 0, p1 + q1 > 0, p0 + q0 > 0 and p2(p1 + q1) – (p0 + q0) > 0. Therefore, by the Routh-Hurwitz criterion, the positive equilibrium E* is locally asymptotically stable.
For τ > 0, if iω(ω > 0) is a solution of (2.13) if and only if ω satisfies
Separating the real and imaginary parts, we have
which implies
where
Obviously, if r < r1 and (H1) hold, then
If r1 < r < min{r2, r4} holds, which implies
Denote
By Theorem 3.4.1 in Kuang [30], we see that if
We now claim that
This shows that there exists at least one eigenvalue with a positive real part for τ > τ0. Moreover, the conditions for the existence of a Hopf bifurcation [31] are then satisfied yielding a periodic solution. To this end, differentiating Eq. (2.13) with respect to τ, it follows that
Hence, a direct calculation shows that
We derive from (2.15) that
Therefore, we yield
Thus, the transversal condition holds and a Hopf bifurcation occurs at ω = ω0, τ = τ0. Now, let us summarize our results as follows:
Theorem 2.4
If (H1) and r < r1 hold, then the positive equilibrium E* of system (1.4) is locally asymptotically stable for all τ ≥ 0.
If (H1) and r1 < r < min{r2, r4} hold, then there exists a τ0 > 0 such that E* is locally asymptotically stable when τ ∈ [0, τ0). Furthermore, system (1.4) undergoes a Hopf bifurcation at E* when τ = τ0.
Remark
Let τ = 0 and m = 0 in (1.4), then Theorem 2.3 of this paper is equivalent to Theorem 3 of [21]. Obviously, we generalize the conclusion of boundary equilibrium in [21], and show that prey refuge affect the stability of the boundary equilibrium. Further, the global stability of the boundary equilibrium will be studied in Section 4.
Notice that the conditions of positive equilibrium E* is locally asymptotically stable in [21] is very complicated. Let τ = 0 and m = 0 in model (1.4), compare Theorem 2.4 of this paper with Lemma 4 of [21], we find that the conditions of our positive equilibrium locally stable are more extensive and concise than that of [21].
3 Permanence
Lemma 3.1
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Assume that (H1) holds, then
where
Proof
It follows from (H1) that
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). From the first equation of system (1.4), it follows that
Applying Lemma 2.3 in [32] to (3.2), it immediately follows that
Then for above ε > 0 sufficiently small there exists a T1 > 0 such that if t > T1, x(t) ≤ L1 + ε. We derive from the second and the third equations of system (1.4) that for t > T1 + τ,
Consider the following auxiliary equations:
Using a similar argument as that in the proof of Lemma 2.4 in [33], it follows from (3.5) that
By comparison, we obtain that
Let ε → 0, it follows that
The proof is complete.
Lemma 3.2
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5), if (H1) and
hold, then
where
Proof
It follows from (H1) and condition (3.9) that
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). For above ε > 0 sufficiently small, it follows from Lemma 3.1 that there exists a T2 > 0 such that if t > T2, y1(t) ≤ L2 + ε, y2(t) ≤ L3 + ε. Hence, we derive from the first equation of system (1.4) that for t > T2,
According to condition (3.9) and similar to the proof of Lemma 3.1, we have
For above ε > 0 sufficiently small, there exists a T3 ≥ T2 such that if t > T3, x(t) ≥ l1 − ε. Therefore, it follows from the second and the third equations of system (1.4) that for t > T3 + τ,
Consider the following auxiliary equations:
Using a similar argument as that in the proof of Lemma 2.4 in [33], it follows from (3.13) that
By comparison, we obtain that
Let ε → 0, we conclude that
The proof is complete.
As a direct corollary of Lemmas 3.1 and 3.2 we have the following theorem.
4 Global stability
In this section, we study the global stability of the predator-extinction equilibrium E1 and the global attractivity of the coexistence equilibrium E* of system (1.4). The strategy of proofs is to use Lyapunov functionals and the LaSalle invariance principle.
Theorem 4.1
If (H2) holds, then the predator-extinction equilibrium E1(r/a, 0, 0) of system (1.4) is globally asymptotically stable.
Proof
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Denote x0 = r/a.
Define
where ξ1 =
Calculating the derivative of V11 along positive solutions of system (1.4), it follows that
Substituting r = ax0 into (4.2), we obtain that
Define
We derive from (4.3) and (4.4) that
If (H2) holds, it then follows from (4.5) that V̇1(t) ≤0. By Theorem 5.3.1 in [34], solutions limit to 𝓜, the largest invariant subset of {V̇1(t) = 0}. Obviously, we see from (4.5) that V̇1(t) = 0 if and only if x = x0, y2 = 0. Noting that 𝓜 is invariant, for each element in 𝓜, we have x(t) = x0, y2(t) = 0. It therefore follows from the third equation of system (1.4) that 0 = ẏ2(t) = βy1(t), which yields y1(t) = 0. Therefore, V̇1(t) = 0 if and only if (x, y1, y2) = (x0, 0, 0). Accordingly, the global asymptotic stability of E1 follows from LaSalle invariance principle. This completes the proof.
Theorem 4.2
Assume that (H1) and (3.9) are satisfied. If the following holds:
where
Proof
Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Define
where
Calculating the derivative of V11 along positive solutions of system (1.4), it follows that
Substituting
Define
It follows from (4.9) and (4.10) that
Noting that
and
we derive from (4.11) that
where
Hence, if (4.6) holds, it then follows from (4.12) that V̇1(t) ≤0, with equality if and only if x = x*,
5 The influence of prey refuge
In this section, we investigate the influence of prey refuge. Under the condition (H1), let us compute the derivative along the positive equilibrium E* with respect to m, that is
Denote
Due to the existence of E*, which implies that x* is a strictly increasing function of m, that is, increasing the constant amount of prey refuge m leads to the increase of prey densities. When 0 < m < 1 − ℜ1,
6 Numerical simulations
Example 6.1
In (1.4), let m = τ = 0, then system (1.4) is degenerated into model (2.1) of [21]. Let r = 6, a = 3, μ = 20, b = d = e = 1, η = β = 2 and c = 4, then system (1.4) has a positive equilibrium E* = (0.5168, 1.5330, 3.0660). By calculation, μη = 40 > 1.5 = be(1 +
Note that ηβ(1 + bx*) = 6.0671 > 0.4778 =
m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) is locally asymptotically stable.
m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) with different initial values is locally asymptotically stable.
Example 6.2
In (1.4), let r = 6, a = 3, m = 0.2, b = d = e = 1, η = β = 2 and c = 4, then be(1 +
If μ = 1, τ = 10, it is easy to show that μη = 2 > be(1 +
Figure 3μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) is globally asymptotically stable.
Figure 4μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) with different initial values is globally asymptotically stable.
If μ = 20, τ = 10, it is easy to show that μη = 40 > be(1 +
Figure 5μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) is locally asymptotically stable.
Figure 6μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) with different initial values is locally asymptotically stable.
If μ = 36, it is easy to show that μη = 72 > be(1 +
If μ = 2, τ = 2, it is easy to show that μη = 4 > be(1 +
Figure 9μ = 2, τ = 2, E* = (1.9487, 0.1998, 0.3996) of system (1.4) with different initial values is globally attractive.
Example 6.3
In (1.4), let r = 10, μ = 6, c = e = 0.5, a = b = d = 1, η = β = 2 and τ = 0. By simple computations, we find μη = 10 > 0.75 = be(1 +
Bifurcation diagram of the prey species of model (1.4), m as the bifurcation parameter.
Bifurcation diagram of the predator species of model (1.4), m as the bifurcation parameter.
τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from zero to 1 − ℜ1 = 0.4432, the predator species will increase.
τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from 1 − ℜ1 = 0.4432 to 0.9919 = 1 − ℜ2, the predator species will decrease.
7 Conclusion
In this paper we investigate the influence of prey refuge on the dynamics of a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay. Sufficient conditions are derived to ensure the predator-extinction and the locally asymptotically stability of positive equilibrium. Compared with [21] we get more precise conditions. Also, we find that time delay can cause a stable equilibrium to become unstable one, even Hopf bifurcation to occur, when time delay passes through some critical values. Furthermore, the persistence is investigated. After that, we study the global stability of the predator-extinction and positive equilibrium by constructing some suitable Lyapunov functionals.
Also, we discuss the influence of prey refuge on the densities of predator species and prey species. When prey refuge m in the interval (0, 1 − ℜ1), the density of predators will increase with prey refuge m, due to predator species having sufficient food for their predation with sufficiently small prey refuge m. Predator population attains its maximum when the prey refuge m = 1 − ℜ1. In case of larger values of m (m > 1 − ℜ1), this implies that predators species are less likely to catch prey, and the predator species deceases with the increasing of prey refuge m. Eventually, the predator population will be extinct when prey refuge m = 1 − ℜ2.
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Competing interests The authors declare that there is no conflict of interests.
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Authors’ Contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (11601085), the Natural Science Foundation of Fujian Province (2017J01400) and the foundation of Education Department of Fujian Province (JAT160063).
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