Abstract
In this paper, a prey-predator model and weak Allee effect in prey growth and its dynamical behaviors are studied in detail. The existence, boundedness, and stability of the equilibria of the model are qualitatively discussed. Bifurcation analysis is also taken into account. After incorporating the searching delay and digestion delay, we establish a delayed predator-prey system with Allee effect. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. Finally, we present some numerical simulations to illustrate our theoretical analysis.
1. Introduction
Some researchers have conducted extensive research on the dynamics of interacting prey-predator models to understand the long-term behavior of species. A wide variety of nonlinear coupled ordinary differential equation models are proposed and analyzed for the interaction between prey and their predators. The classic predator-prey model is the Lotka-Volterra model, which was independently proposed by Lotka in the United States in 1925 and Volterra in Italy in 1926 [1, 2]. The model was developed on the basis of a single-population growth model and has wide applicability. The mathematical form of the Lotka-Volterra model isIn population dynamics,when the population density is very low, there is a positive correlation between the population unit growth rate and the population density. This phenomenon can be called the Allee effect [3–5], starting with Allee’s research [6]. The Allee effect is classified according to the density-dependent properties at low density. If the population density is low, a strong Allee effect will appear. If the proliferation rate is positive and increases, the Allee effect will be weak. Demographic Allee effects can be either weak or strong [7, 8]. When the density is below the critical threshold, the population affected by the strong Allee effect will have a negative average growth rate. Under deterministic dynamics, we find that populations that do not exceed this threshold will be extinct. Many jobs only consider the strong Allee effect, but in the work of Allee it is clear that the Allee effect also has a weak Allee effect [8–13].
Today, it is widely believed that the Allee effect greatly increases the likelihood of local and global extinction and can produce a rich variety of dynamic effects [14–16]. And it is interesting and important to study the impact of Allee effect on the predator-prey models [17–19]. In this paper, we introduced a predator-prey model with weak Allee effect:Here, the weak Allee effect term is , where is described as a “weak Allee effect constant” ([12]). is the prey population and is the predator population, is the intrinsic death rate of predators, is the conversion efficiency from prey to predator, is the carrying capacity, is the intrinsic growth rate of prey, and is the prey capture rate by their predators. It is more realistic to introduce time delay on the basis of traditional predator-prey model because it exists almost everywhere in biological activities and is considered as one of the reasons for the regular change of population density [20–26]. Therefore, in order to make the system established in this paper biologically closer to reality, incorporating the searching delay and digestion delay in the system (2) is interesting. Based on the above considerations, We establish a predator-prey model with time delay and weak Allee effect, as follows:where the time delay is the controlling or perturbed parameters, is the searching delay, and is the digestion delay.
The latter parts of the paper are described as follows. In Section 2, we discuss the boundedness, the stability of the equilibria, and bifurcation of the model (2) in detail. In Section 3, we investigated local stability property of interior equilibrium point of the model (3) with time delay; the Hopf bifurcation around the positive equilibrium point is also studied. In Section 4, we verify the previous theoretical derivation by numerical simulation.
2. A Predator-Prey Model with Weak Allee Effect
We easily see that model (2) exhibits three equilibrium points , , and . Here , . And for the positive equilibrium point(s), we have .
2.1. Boundedness
Theorem 1. For the solution of model,
Proof. We define . Then we can easily see that along the solution of system (2),Thus, we see that for all large Hence the standard comparison argument shows that
2.2. Stability Analysis
Theorem 2. (1) Trivial equilibrium point is always a saddle-node point.
(2) is stable for and is a saddle point otherwise.
(3) Coexistence equilibrium is locally asymptotically stable for and is unstable node otherwise.
Proof. Let So, the Jacobian matrix for the model (2) is given by whereSo we get First, it can be concluded by calculating the Jacobian matrix of the model (2) at given by And hence is always a saddle-node point. Then, by evaluating the Jacobian matrix of the model (2) at , we find First eigenvalue is negative; hence is stable if implying , and is a saddle point when . Finally, the Jacobian matrix for the model (2) evaluated at is given by The characteristic polynomial is where and . Thus, we have the following conclusions. (1) If and , then the positive equilibrium is locally asymptotically stable. (2) If and , then the positive equilibrium is unstable.
Theorem 3. is globally stable when .
Proof. Consider the Lyapunov function: The derivative of along the solution of the model is
2.3. Bifurcation Analysis
2.3.1. Transcritical Bifurcation
Theorem 4. The model enters into transcritical bifurcation around at , where .
Proof. One of the eigenvalues of will be zero if which gives . At this point, the other eigenvalue is . If and denote the eigenvectors corresponding to the eigenvalue 0 of the matrices and , respectively, then we obtain and , where , , .Therefore, by the Sotomayor theorem, we can find that the model experiences transcritical bifurcation at around the axial equilibrium .
2.3.2. Hopf Bifurcation
From Theorem 2, model (2) undergoes bifurcation if . The purpose of this section is to show that model (2) undergoes a Hopf bifurcation if . We analyze the Hopf bifurcation occurring at by choosing as the bifurcation parameter. DenoteWhen , we have . Thus, the Jacobian matrix has a pair of imaginary eigenvalues . Let be the roots of ; thenandBy the Poincare-Andronov-Hopf Bifurcation Theorem, we know that model (2) undergoes a Hopf bifurcation at when . However, the detailed nature of the Hopf Bifurcation needs further analysis of the normal form of the model. Set and , to as origin of coordinates . We have the following model:where , , , , andwhere and are smooth functions of and at least of order four. Now, using the transformation, , we obtainwheresoSetwhereSoIf , the equilibrium is destabilized through a Hopf bifurcation that is supercritical and Hopf bifurcation is subcritical otherwise.
3. Delayed Model with Weak Allee Effect
Let , ; then the model (3) can be expressed as in the following matrix form after linearization:
3.1. Stability Analysis
The characteristic polynomial iswhereLetSoLet , , and .
Theorem 5. Assume , when ; we have the following conclusions. (1) When and , the positive equilibrium is locally asymptotically stable. (2) Hopf bifurcation occurs when passes the critical value
Proof. The characteristic equation is Next we suppose that is a solution of for some ; then we have Then Where , , we know So we get Separate real and imaginary partsThenSo We assume thatIf and , then all roots of equation have negative real parts for all ; that is, the equilibrium is locally asymptotically stable: where If , and , there is a unique positive solution; the equilibrium is unstable. Also if , , and , then there are two positive solutions. We have So Then, we get It shows that if and , has a pair of imaginary eigenvalues ,
when .
Next verify the cross-sectional conditions: According to At this time, , where is the value of at . We get Then
Theorem 6. Assume , when ; we have the following conclusions. (1) If , the positive equilibrium is locally asymptotically stable. (2) When , if , the positive equilibrium is locally asymptotically stable; if , the positive equilibrium is unstable. (3) Hopf bifurcation occurs when passes the critical value
Proof. The characteristic equation is Next we suppose that is a solution of for some ; then we have Then Where , , we know So we get Separate real and imaginary partsThenSo We assume thatWe can easily find If , the positive equilibrium is locally asymptotically stable,, and the positive equilibrium is unstable. We have So Then, we get It shows that if, has a pair of imaginary eigenvalues ,
when .
Next verify the cross-sectional conditions: According to At this time,, where is the value of at . We get Then
Theorem 7. Assume , when ; we have the following conclusions. (1) When and and , the positive equilibrium is locally asymptotically stable. (2) Hopf bifurcation occurs when passes the critical value
Proof. The characteristic equation is Next we suppose that is a solution of for some ; then we have Where , , we know So we get Separate real and imaginary partsLet . ThenSo We assume thatIf and , then all roots of equation have negative real parts for all ; that is, the equilibrium is locally asymptotically stable: where If , , and , there is a unique positive solution; the equilibrium is unstable. Also if , , and , then there are two positive solutions. We haveSoThen, we get It shows that if , has a pair of imaginary eigenvalues ,
when .
Next verify the cross-sectional conditions: According to At this time,, where is the value of at . We get Then
4. Numerical Simulations
In this section, we present some numerical simulations to illustrate our theoretical analysis.
First, we theoretically analyze a predator-prey model with Allee effect in the article and obtain the stability conditions of and . Secondly, we carry out numerical simulation and select appropriate the parameters, which draw the stable positions of the equilibrium points and , shown in Figures 1 and 2. In the analysis later in this chapter, we use as the bifurcation parameter to obtain the critical value of the Hopf bifurcation generated by the model (2). By comparing Figures 3 and 4, we find that as the parameter changes the equilibrium point changes from a steady state to a limit cycle. To further verify our point of view, we have made a bifurcation diagram as shown in Figure 5. We found that the bifurcation parameter produced a bifurcation at about 100, which coincided with our previous guess. By the same analysis method, we give a set of timing diagrams for comparison as shown in Figures 6 and 7; we found that with the increase of the parameter the model (2) is shown in Figure 6 and the population gradually becomes stable with the increase of time, while Figure 7 is a periodic change.
Next, we performed a numerical simulation of the model (3), mainly to study the effect of the time-delay parameter on the stability of the coexistence equilibrium point. Here, we compare three sets of timing diagrams, which are the effect of on the stability of the coexistence equilibrium point, the influence of on the stability of the coexistence equilibrium point, and the influence of on the stability of the coexistence equilibrium point. First, by comparing Figures 8 and 9, we find that when , the population gradually becomes stable with the increase of time. When , the population gradually shows periodicity with the increase of time. By comparing Figures 10 and 11, we find that when , the population gradually becomes stable with the increase of time. When , the population gradually shows periodicity with the increase of time; by comparing Figures 12 and 13, we found that when , the population gradually becomes stable with the increase of time. When , the population gradually shows periodicity with the increase of time. At the same time, we selected the appropriate parameters and gave three sets of phase diagrams for comparison. By comparing Figures 14 and 15, we find that when increases from to , the coexistence equilibrium point changes from a steady state to a limit cycle; by comparing Figures 16 and 17, we find that is increased from to . At this time, the coexistence equilibrium point changes from a steady state to a limit cycle; by comparing Figures 18 and 19, we find that when increases from to , the coexistence equilibrium point changes from a steady state to a limit cycle. From this, we can conclude that the stability of the equilibrium point of the model (3) changes with the increase of the time lag when the time-delay parameter is introduced, and the generation of the limit cycle (periodic solution) is accompanied by this change.
5. Conclusions
In this paper, we establish a predator-prey model with a weak Allee effect and demonstrate and analyze the boundedness and stability of the model. We also prove that the model (2) experiences transcritical bifurcation around the axial equilibrium and the model (2) undergoes a Hopf bifurcation at when ; we also analyzed the direction and stability of Hopf bifurcation. Immediately after we introduced the searching delay and digestion delay in the model (2), a new model was obtained, and the model (3) was analyzed for stability changes caused by time lag. It is concluded that the stability of the coexistence equilibrium point of the model (3) changes as the time lag increases. Finally, we verify our theoretical derivation by numerical simulation. First, we select the appropriate parameters to satisfy the stable conditions that we deduced in the article and obtain the stable phase diagrams of the equilibrium points and . Second, we try to change the value of parameter , the timing diagram, phase diagram, and bifurcation diagram corresponding to each parameter drawn. Further verification of our conclusion is as increases, the model (2) will produce bifurcation. We also carry out a numerical model of the model (3) and discuss our numerical results in three groups: at the beginning, let change, ; we observe the timing diagram corresponding to the model (3) as increases. In the change of the phase diagram, we find that with the increase of the coexistence equilibrium point of the model (3) begins to stabilize and becomes unstable and is also accompanied by the generation of the limit cycle (periodic solution); we also change the , ; we observe the increase of , corresponding to the change of the phase diagram of the time series of the model (3); we find that with the increase of the coexistence equilibrium point of the model (3) begins to stabilize and becomes unstable. At the same time, it is accompanied by the generation of the limit cycle (periodic solution); the same we let , so that increases; we also found similar changes. In this process, we also found an interesting phenomenon. When takes a small value, the stability of the coexistence equilibrium point of the model changes. However, requires a larger value than . When , only needs to take a small value, and the stability of the coexistence equilibrium point of the model changes. From a biological point of view, we find that the introduction of weak Allee effect will change the stability of the model; that is to say, the stable state between populations will be broken. Similarly, delays can destroy the stability of the original predator-prey model. Moreover, the introduction of delay and weak Allee effect makes the model closer to reality and makes us more accurately understand the dynamic changes of interspecific relationships. Therefore, we can find that Allee effect and delay play an important role in describing population dynamics.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (31260098, 31560127), the Fundamental Research Funds for the Central Universities (31920180116, 31920180044, and 31920170072), the Program for Yong Talent of State Ethnic Affairs Commission of China (No. 2014121), Gansu Provincial First-Class Discipline Program of Northwest Minzu University, and Central Universities Fundamental Research Funds for the Graduate Students of Northwest Minzu University (Yxm2019109).