Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

1.   Introduction

In recent years, predator-prey systems with integrated pest control have been extensively studied to realize the effective control of pests ^[1,2,3,4]. Generally, IPM (Integrated Pest Management) mainly uses chemical control, biological control and artificial control to establish an efficient comprehensive system to control pests. Among them, biological control is mainly the use of natural enemies, which involves their harvesting and capture pests, while chemical control is the use of pesticides and other chemicals to quickly kill pests in large quantities. Regarding implementing pest control, the combination of two strategies can yield an optimal control strategy ^[5,6].

The functional response is defined as the number of prey killed by a predator per unit of time, and it describes the amount of biological transfer between different trophic levels, i.e., the effects of the predator population on the prey. Many scholars have considered predator-prey models with different functional responses from many aspects. For example, the nonlinear functional response first appeared in the Lotka-Volterra ^[7] predation system. Also, Wang et al. ^[8] and Cheng et al. ^[9] considered Holling-Ⅰ functional responses in a predator-prey model; Luo et al. ^[10] and Liu et al. ^[11] focused on Holling type Ⅱ functional responses. Other scholars ^[12,13] studied the Holling-Ⅲ functional response; through their research, it was found that functional response terms not only rely on prey behavior, but they are also related to predator activity. Wang ^[14] researched the predator-prey model with ratio-dependent functional response terms, and by applying a ratio-dependent functional response, it was found that the predator density could not be zero, i.e., the predator and prey could not be extinct at the same time. Thus, the Beddington-DeAngelis functional response term ^[15,16] has garnered much attention.

To facilitate research, many scholars have set the periodic release of natural enemies in the pest-natural enemy system as an invariant constant ^[17]; in fact, pest outbreaks are generally transient, uncertain and unpredictable, so pest control strategies incorporate pulse disturbances to simulate the effects of the sudden change of actual factors on the state of the system. Moreover, many anthropogenic phenomena in nature can be described by impulsive differential equations, and these phenomena include fishing and stocking in fish farming, spraying pesticides in agriculture the regular release of natural enemies, insulin injection in diabetes, etc. In ^[18,19], the authors considered a predator-prey model with linear pulses, employing modified versions of the Leslie-Gower and Holling-Ⅱ functional responses, but these impulse effects tend to be nonlinear. However, the excessive use of pesticides or biological predators can potentially yield significant damage to the environment and economy. Nonetheless, optimal impulse control management requires that the IPM rely on the densities of both pests and natural enemies in the field; therefore, predator-prey models with nonlinear impulse control are more practical and reasonable. To investigate how nonlinear impulse control affects the dynamic behavior of a system, in ^[20], Li et al. studied a predator-prey model with nonlinear pulses and a Beddington-DeAngelis functional response, finding that the models with nonlinear impulse control exhibit richer dynamic behavior than those with linear pulses. The Leslie-Gower functional response is an important functional response, which means that the predator population does not unlimitedly grow because of the limitations of natural resources, and many predators do not live on just one kind of food. The authors of ^[21,22] studied the dynamic behavior of predatory models with Leslie-Gower and Holling-Ⅱ terms. However, these studies applied the assumption that the impulse control is linear or constant. Because the outbreak of pests and diseases occurs rapidly, it is difficult for many countries or regions, which have limited capacity for agricultural resources such as pesticides, labor, and biological resources, to effectively control an outbreak of pests. In order to take the resource limitation into account, the instant killing rate should be a monotonically increasing saturation function, and the natural enemies that are released should be based on the current pest and prey densities in the field. With the goal of realizing optimal pulse management, we studied a nonlinear impulsive control predator-prey model with a Leslie-Gower term and Beddington-DeAngelis term, which more comprehensively consider the effects of human control and environmental factors on population density:

where $x(t)$ and $y(t)$ are the populations of pests and natural enemies at time $t$, respectively $\alpha, \beta, a_1, a_2, a_3$, $k, h, q$, $\theta$, $\delta$, $w_1, w_2$ are model parameters that assume only positive values, $r_1, r_2$ represent the birth rates of pests and natural enemies, $b$ describes the intraspecific competition for pests, $\delta$ measures the maximum rate at which pests are killed by spraying insecticides ($0 < \delta\leq1$), $h$ is the semi-saturation constant, $w_1, w_2$ denote the maximum releases of predators based on prey density and predator population density, and $\frac{{\alpha x(t)y(t)}}{{{a_1} + {a_2}x(t) + {a_3}y(t)}}$ is the B-D functional response term; IPM strategies are adopted at each discrete time point.

2.   Global stability of periodic solutions

First, we give some notations, definitions and lemmas which will be useful for our main results.

Assume that

are continuous on $(nT, (n + 1)T]$ and $f = {({f_1}, {f_2})^T}$ is the right-hand mapping of the first two equations of the system (1.1); obviously, the smoothness of $f$ guarantees the existence of a unique solution. The following lemma ^[23,24] is given.

Lemma 2.1. Let $X(t) = (x(t), y(t))$ be a solution of system (1.1), and if the initial value $X({0^ + }) \ge 0$, then $X(t) \ge 0$ for all $t \ge 0$; furthermore, if $X({0^ + }) > 0$, then $X(t) > 0$ for all $t \ge 0$.

Proof. Obviously for $y(t)$, through scaling we have the following:

It is easy to see that

for $t \in (0, T]$.

At time $t = T$, $y({T^ + }) = qy(T) + {w_2} > qy(T) > 0$; thus, when $t \in (T, 2T]$, the initial value becomes $y({T^ + }) \ge 0$. For $x(t)$, through scaling we get the following:

Thus, we have

for $t \in (0, T]$.

At time $t = T$, $x({T^ + }) > (1 - \delta)x(T) > 0$; thus, when $t \in (T, 2T]$, the initial value becomes $x({T^ + }) \ge 0$.

Similar to the previous proof we can determine when $X({0^ + }) \ge 0$ and $X(t) \ge 0$ for all $t \ge 0$. Similarly, when $X({0^ + }) > 0$ we can get that $X(t) > 0$ for all $t > 0$. □

Lemma 2.2. Define $g:{R_ + } \times {R_ + } \to R$ as continuous on $(nT, (n + 1)T]$ when $n \in {Z_ + }, u \in {R_ + }$, and

${\psi _n}:{R_ + } \to {R_ + }$ is a non-decreasing mapping; suppose that the following inequalities hold:

and that the impulsive differential equations given by

have a maximum solution $r(t) \in [0, + \infty)$; when $t \ge 0$ and $V({0^ + }, {x_0}) \le {u_0}$, we have that $V(t, x) \le r(t)$.

The elimination of pest populations is an important goal of IPM strategies; therefore, studying the existence and global stability of periodic solutions plays a crucial role in the investigation of their dynamic behavior. Thus, we consider the subsystem of (1.1):

Now, we shall give the basic properties of the subsystem:

Theorem 2.3. If $1 - q\exp ({r_2}T) < 0$, system (2.5) has a globally asymptotically stable periodic solution

where ${y^*} = A + \sqrt {{A^2} + B}$ with $A = \frac{{{w_2}}}{2} + \frac{{{r_2}(1 - q{e^{{r_2}T}})}}{{2\frac{\beta }{k}(1 - {e^{{r_2}T}})}}$ and $B = \frac{{{r_2}{w_2}}}{{ - \frac{\beta }{k}(1 - {e^{{r_2}T}})}}$ as two positive constants.

Proof. By applying the idea of variable separation and difference equations to solve the system (2.5), we have

and $y((n + 1){T^ + }) = qy((n + 1)T) + {w_2}$; then, the following equation is obtained

Let $y(n{T^ + }) = y((n + 1){T^ + }) = {y^*}$. Substituting ${y^*}$ into the above equation yields the following quadratic equation:

We have that ${y^*} = A + \sqrt {{A^2} + B}$, where

Apparently when $1 - q{e^{{r_2}T}} < 0$, system (2.5) has a periodic solution $y^*(t)$ for the case with pests.

According to the above analysis, we can easily get the expression for the unique pest-free periodic solution of system (1.1), i.e., $(0, {y^*}(t))$. In what follows, we will give the sufficient condition for the global stability of the pest-free periodic solution $(0, {y^*}(t))$ of system (1.1). □

Theorem 2.4. The periodic solution $(0, {y^*}(t))$ of the system (1.1) is locally asymptotically stable for the case with pests if

Proof. Here we first demonstrate the local stability of the periodic solution for pests.

The local stability of the periodic solution for pests may be determined by considering the behavior of small-amplitude perturbations $(u(t), v(t))$ of the solution. Define $u(t) = x(t), v(t) = y(t) - {y^*}(t)$, where $u(t), v(t)$ are small perturbations. After neglecting the higher-order terms, it can be written as

where $\varphi(t)$ satisfies

and $\varphi(0) = I$ is the identity matrix. Hence, the fundamental solution matrix is given by

Then, linearizing the third and fourth equation of system (1.1), we have

Let

If each eigenvalue of $M$ has an absolute value that is less than one, the solution is locally stable; note that (*) is not important, so it is not considered. Below we define

as two eigenvalues of the matrix $M$ because

It is easy to see that ${\lambda _2} < 1$, according to the Floquet theory for impulsive differential equations, $(0, y^{*}(t))$ is locally stable if

Next, we will prove the global attraction of the periodic solution for pests. Just prove that the arbitrary solution $(x(t), y(t))$ tends to $(0, y^{*}(t))$ when $t \to \infty$.

(1) Prove that $x(t)\to0$.

Since $x = \frac{{{r_1}}}{b}$, it follows that $\frac{dx}{dt} < 0$; then $0 < x < \frac{r_1}{b}$. Take a small enough $\varepsilon > 0$ so that

It follows from the system (1.1) that

further, we consider the following impulsive differential equation:

By applying the impulsive differential equation theorem and the comparison theorem, within the interval $((nT), (n+1)T)$, we get

Then, $x(nT) \le x({0^ + }){\eta ^n}$ and $x(nT) \to 0$ as $n \to \infty$. Therefore $x(t)\to 0$ as $t\to0$.

(2) Prove that $y(t)\to {y^*}(t)$.

From above $x(t)\to 0$ as $t\to\infty$ for any ${\varepsilon _1} > 0$. Additionally, there must exist a $T_1 > 0$, such that $0 < x(t) < {\varepsilon _1}$ and $t > T_1$. Without loss of generality, we may assume that $0 < x(t) < {\varepsilon_1}$ for all $t > 0$; from system (2.5) we have

Then we consider the following equations

and

where

and

For arbitrarily small ${\varepsilon_2} > 0$, according to the comparison theorem we have

Let ${\varepsilon_1\to 0}$ as $t\to\infty$; we have that ${y^*}(t) - {\varepsilon _2} < y(t) < {y^*}(t) + {\varepsilon_2}$; hence, $y(t)\to y^*(t)$ for $t$ large enough. Thus the proof is completed. □

3.   Permanence

In order to understand the long-term survival of the prey and predator populations, it is important to investigate the permanence of system (1.1). First, we give the definition of permanence as follows:

Definition 3.1. The system (1.1) is permanent if there exists $M \ge m > 0$ for any solution $(x(t), y(t))$ of (1.1) with ${x_0} > 0$ such that

Theorem 3.2. System (1.1) is permanent if

Proof. Suppose that $(x(t), y(t))$ is any solution of system (1.1). Apparently, given the above analysis, we have that $x < M_1 < \frac{r_1}{b}$ for $t$ large enough. For convenience, let $x(t)\le M$ for any $t > 0$. Then, from Theorem 2.4, we can easily obtain that $y^{*}(t)-{\varepsilon_2} < y(t) < y^{*}(t)+{\varepsilon_2}$ for $t$ large enough. Let $m_2 = y^{*}(t)-{\varepsilon_2}$ and $M_2 = y^{*}(t)+{\varepsilon_2}$; then we have that $m_2\le y(t)\le M_2$.

Assume that $y(t) \ge {m_2}$ for any $t > 0$. We need to prove that there exists a positive constant $m_1$ such that $x(t)\ge {m_1}$ when $t$ is large enough. We will prove it in two steps:

1) Take a positive number $m_3 > 0$ and $\varepsilon_3 > 0$ arbitrarily small enough. Let

where

and

Next we will prove that there must exist a $t_3 > 0$. Set $x(t_3)\ge {m_3}$ for $t > t_3$. Otherwise, from Theorem 2.4, we obtain that $y(t)\le {z_3(t) and z_3(t)\to {z^*}_3(t)}$ as $t\to \infty$, where $z_{3}(t)$ is the solution of system (3.4).

and

Thus there is a $T_1$ such that $y(t)\le{z_3(t)}\le{{z^*}_3(t)+\varepsilon_3}$ as $t > T_1$. Therefore we establish (3.6) as below:

Let $N \in {Z_ + }, NT > T$; integrating (3.6) on $(nT, (n+1)T)$, $n\ge{N}$, we have

Then, given (3.3), substituting the right-hand side of (3.7) with $\eta_{2}$, we get that $x((n+1)T)\geq x(nT)\eta_{2}$. Then $x((N+n)T) > x(nT)\eta_2^n\to\infty$ as $n\to\infty$, which is a contradiction; hence, there must exist a $t_{3} > 0$ such that $x(t_3)\ge{m_3}$.

2) Assume that $x(t)\ge{m_3}$ for all $t > t_3$ and prove that the conclusion holds; otherwise, let $t^{*} = \inf{t > t_3, x(t) < m_3}$, and we have that $x(t)\ge {m_3}$ as $t\in[t_3, t^{*}]$. It is divided into the following two cases: points in time of impulse and no impulse.

Case (1): ${t^*} = nT, {n_1} \in {Z_ + }$. We have that ${q_1}{m_1} \le {q_1}x({t^*}) = x({t^*}^ +) \le {m_3}$. Select $n_2, n_3\in{Z_+}$ such that

where $\rho = {r_1} - b{m_3} - \frac{\alpha }{{{a_1}}}{M_2} < 0$. Let ${T'} = {n_2}T + {n_3}T$; there must exist a ${t_4} \in ({t^*}, {t^*} + {T'})$ such that $x(t_4)\ge{m_3}$. Otherwise, assume that $x(t) < {m_3}$ for any $t \in{ (t^{*}, t^{*} + T')}$; since (3.4) has an initial solution ${z_3}({t^{* + }}) = y({t^{* + }})$, we can get

According to the above, we know that

Since $y(t)\le M_2$, when $t \in [{t^*}, {t^*} + {n_2}T]$, we have

Integrating (3.8) on the interval $[{t^*}, {t^*} + {n_2}T]$, we obtain

So,

which contradicts the above assumption and means that there must exist a ${t_4} \in ({t^*}, {t^*} + {T'})$ such that $x({t_4}) > {m_3}$.

Let $\bar t = \inf \{ t > {t^*}, x(t) \ge {m_3}\}$; we have that $x(t) < {m_3}$ when $t \in \left({{t^{*}}, \bar t} \right]$ and $x(\bar t) = {m_3}$. Assume that $\bar t = \inf \{ t > {t^*}, x(t) \ge {m_3}\}$; from (3.8), it follows that

Therefore we get that $x(t) \ge {m_1}'$ for $t \in \left({{t^*}, \bar t} \right]$ when $t > \bar t$. We can still prove that the conclusion $x(\bar t) \ge {m_3}$ holds by employing the same method as above.

Case (2): ${t^*}\ne nT, n \in {Z^ + }$. Then $x({t}) \geq {m_3}$ for ${t}\in [{t_3}, {t^*})$ and $x(t^*) = {m_3}$. Assume that

There are two possible cases for $x(t)$.

(H1): If $x(t) < {m_3}$ for all $t \in ({t^*}(n_1' + 1)T]$, it is similar to Case (1). We can prove that there exists a ${t_5} \in \left[ {(n_1' + 1)T, (n_1' + 1)T + {T'}} \right]$ such that $x({t_5}) \ge {m_3}$. Otherwise assume that $x(t) < {m_3}$ for all $t \in \left[ {(n_1' + 1)T, (n_1' + 1) + {T'}} \right]$. Consider the initial solution ${z_3}((n_1' + 1){T^*}) = y((n_1' + 1){T^*})$ of (3.4) with the initial values on $\left[ {{t^*}, (n_1' + 1 + {n_2} + {n_3})T + {T'}} \right]$; we get that

Integrating (3.8), we have

Therefore,

This contradicts the assumption.

Let $\hat t = \inf \left\{ {t > {t^*}, x(t) > {m_3}} \right\}$. We have that $x(t) < {m_3}$ when $t \in ({t^*}, \hat t)$ and $x(\hat t) = {m_3}$. For $t \in ({t^*}, \hat t)$, suppose that

We have

Thus, $x(\hat t) \ge {m_3}$ for $t > \hat t$. By repeating the above steps we can obtain that $x(t) \ge {m_1}$ as $t \ge {t_3}$.

(H2): If there exists a $t \in ({t^*}, (n_1' + 1)T)$, we have that $x(t) \ge {m_3}$. Let ${t'} = \inf \left\{ {t > {t^*}, x(t) > {m_3}} \right\}$ such that $x(t) < {m_3}, t \in \left[ {{t^*}, {t'}} \right)$, but $x({t'}) = {m_3}$ when $t \in \left[ {{t^*}, {t'}} \right)$; thus, we get

since $x(\hat t) \ge {m_3}$. By repeating the steps above we can obtain that $x(t) \ge {m_1}$ as $t > {t_3}$. In summary, when $t \ge {t_3}$, we can prove that $x(t) \ge {m_3}$. □

4.   Bifurcation

Next, we will research the existence of a nontrivial solution near the pest-eradication solution by applying the bifurcation theorem of Rabinowitz.

Theorem 4.1. System (1.1) has a positive nontrivial periodic solution when $T = T^{*}$, and it is supercritical if $\alpha {a_2} \le 4b{a_1}{a_3}$.

Proof. It is computationally convenient to exchange $x$ and $y$ and change the period $T$ to $\tau$.

let $\Phi$ be the solution of the non-impulsive system associated with system (4.1); also, we get that $X(t) = \Phi (t, {X_0})$ with ${X_0} = X(0)$. Define the mappings ${\Theta _1}, {\Theta _2}:{R^2} \to {R^2}$ by

and the mappings ${F_1}, {F_2}:{R^2} \to {R^2}$ by

Thus, by computation, we get

where ${\tau _0}$ is the root of ${d_0}' = 0$. In fact, it is easy to see that ${d_0}'$ is equivalent to ${\tau _0} = T^{*}$.

By the same method, we have

so we can get

To determine the sign of $B$ and $C$, first assume that

which implies that

Then, $\rho (t)$ is a strictly increasing function. Given that ${d_0}' = 0$, since

it is obvious that $\rho ({\tau _0}) > 0$; so, $\frac{{{\partial ^2}{\Phi _2}({\tau _0}, {X_0})}}{{\partial y\partial {\tau _0}}} > 0$ and $B < 0$. Through analysis, we define the following:

Apparently $- 2b + \frac{{2\alpha {a_2}t}}{{{{({a_1} + {a_3}t)}^2}}} \le 0$ only such that $\alpha {a_2} \le 4b{a_1}{a_3}$. This means that

for all $y^{*}(t) > 0$. So we get

From (4.7) we have that $C > 0$.

In summary, $BC < 0$, and there is a supercritical bifurcation when $T = T^{*}$. □

5.   Numerical simulation

In order to confirm our theoretical results and facilitate interpretation, we will focus on the complex dynamics. The properties of dynamic systems can be obtained by conducting bifurcation analysis.

First, the influence of impulse period $T$ on the dynamic characteristics of the system was studied. From Figure 1, we can ascertain that dynamic behaviors of system (1.1) become increasingly complicated as parameter $T$ increases, and these include behaviors such as period-doubling bifurcation, period-halving bifurcation, chaotic bands, meanwhile, multiple attractors may coexist with the same $T$. Figure 2 shows that the pest free periodic solution has global stability when the impulse period stays below some threshold value in (A, B), and that different attractors can coexist with the same $T = 2.5$ when we select different initial values i.e., $({x_0}, {y_0}) = (1.3, 2.1), ({x_0}, {y_0}) = (3.5, 6.8)$ in (C, D, E, F). The different solutions coexist for a wide range of parameters, which suggests that the final steady state of pest and predator populations depends on their initial densities, and these results confirm that changes in the impulse period $T$ can greatly alter the dynamics of the system (1.1).

It can be ascertained from Figure 3 that the nonlinear impulse parameter $w_1$ affect the dynamics of the system (1.1), and that with the increase of parameter $w_1$, the system (1.1) undergoes period-doubling bifurcation, chaotic bifurcation, period-halving bifurcation and multi-stability. Figure 4 shows that attractors with different amplitudes appeared at $w_1 = 10.2$. Therefore, the initial values of both pest and predator populations are critical.

The results shown in Figure 5 reveal how the maximal release of predators, denoted by $w_2$ affects the dynamics of system (1.1). The properties of the positive periodic solution are maintained until $w_2 = 2.9$, at which point a period-doubling bifurcation occurs. Additionally, as $w_2$ increases, a series of period bifurcations cause the system (1.1) to move from periodicity to chaos. However, when $w_2\geq 20.2$, the chaos disappears.

6.   Conclusions

Regarding the predator-prey systems with an integrated pest management scheme, the impulse control strategies in many previous studies were linear, while ignoring the influence of nonlinear impulse on pest control such as the effects of limited resources. The number of natural enemies to be released must be based on the densities of the pest and predator populations in the field, so we have provided a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D terms. The global stability of the pest extinction periodic solution of system (1.1) has been studied by applying the Floquet theory for differential equations and the comparison theorem, and the conditions for the permanence of system (1.1) has been obtained in detail. We have also proven that a nontrivial periodic solution can be obtained via a supercritical bifurcation once a threshold condition is reached within the context of bifurcation theory. In order to reveal how nonlinear impulses affect successful pest control strategies, we performed numerical simulations to show that system (1.1) has very complex dynamic behaviors, such as period-halving bifurcations, period-doubling bifurcation, chaos, and non-unique attractors. Comparing the bifurcation diagrams for parameters ${w_1}$ and ${w_2}$, it was found that small perturbations can cause periodic oscillations of different amplitudes and periods in the populations of pests and natural enemies, that is, the influence of nonlinear impulses on the dynamic behavior of the model is very sensitive, which shows that the predator-prey models with nonlinear impulses have richer dynamic properties. These bifurcation diagrams show that impulse period, insecticide dose, and initial population densities are critical to successful control strategies.

In view of these results, we believe that systems with nonlinear impulsive control are more reasonable and have more complex dynamic behavior than systems with linear impulsive control. Furthermore, nonlinear impulsive control should be taken into account when investigating predator-prey systems with IPM. However, a more realistic case is that the density dependent releasing function should depend on the densities of the prey and predator populations, i.e., the higher the density of pests or the lower the density of predators in the field, the higher the number of predators that should be released and vice versa. We will work on this in the near future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Nos. 11901370, 12001425); Natural Science Basic Research Plan in Shaanxi Province of China (No. 2020JM-569); Shaanxi Province Natural Science Basic Research Program (No. 2023JCYB258); Shaanxi Ecospatial Governance Key Research Project (No. 2022HZ1809); and Key Project of Postgraduate Education Reform of Xi'an Technological University (No. XAGDYJ220106).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.