Periodic Solution of a Neutral Delay Leslie Predator-Prey Model and the Effect of Random Perturbation on the Smith Growth Model

Li, Tongtong; Zhao, Wencai

doi:https://doi.org/10.1155/2020/8428269

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Abstract Introduction Preliminaries Conclusion and Numerical Simulation Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Dynamical Analysis of Biological Systems

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Research Article | Open Access

Volume 2020 | Article ID 8428269 | https://doi.org/10.1155/2020/8428269

Periodic Solution of a Neutral Delay Leslie Predator-Prey Model and the Effect of Random Perturbation on the Smith Growth Model

Tongtong Li¹and Wencai Zhao¹

Guest Editor: George V. Popescu

Received18 Oct 2019

Revised20 Feb 2020

Accepted19 Mar 2020

Published14 Apr 2020

Abstract

This paper puts forward a class of ratio-dependent Leslie predator-prey models. Firstly, a neutral delay predator-prey model with ratio dependence and impulse control is established and the existence of positive periodic solutions is proved by the coincidence degree theory. Secondly, a stochastic disturbance Leslie model of Smith growth is obtained when the interference of white noise is taken into consideration and the impact of delay is ignored. Applying It’s formula, we get the conditions of system persistence and extinction. Finally we verify the correctness of theoretical analysis with numerical simulations.

1. Introduction

In the population dynamic systems, the relationship between population growth rate and population density is complex and diverse. The Verhulst–Pearl logistic equation [1]is widely used to describe population growth for its concise mathematical expression and clear biological meaning, where denotes the population density at time , is the intrinsic growth rate, and is the environmental capacity. However, in 1963, Smith [2] studied the population dynamics of Daphnia magna and found that the growth of Daphnia magna population in the laboratory did not satisfy the above growth model, and he proposed the following growth equation:which is usually called the Smith growth equation, where is a positive constant. Then, in 1991, Kuang [3] considered a neutral predator-prey model with time delays and the Smith growth as follows:where and are the recovery time delay of the prey population and the digestion time delay of the predator population, respectively, is the death rate of the predator, and is the conversion factor of prey into the predator. is the neutral delay term. indicates the functional response of predator to prey and depends only on the density of the prey population. However, when the predators have to search for food, there will be competition among individual predators for limited prey. The amount of food the predator obtains depends not only on the density of the prey population but also on the density of the predator itself, which is influenced by the relative ratio of the number of predators and prey populations, reflecting the ratio-dependent characteristics. Leslie proposed the following Leslie model [4, 5] in studying the relationship between the predator and prey:where stand for prey and predator densities at time , respectively. and are the intrinsic growth rates of the prey and predator, denotes the density-dependent coefficient of the prey, the parameter is the capturing rate, and is the conversion factor of prey into predators. Motivated by the works [3–5], we propose a ratio-dependent neutral delay Leslie predator-prey model as follows:where all of the parameters are positive and have the biological meanings listed in Table 1.

The first equation of the system states that the growth rate of prey depends not only on its own density but also on the neutral term. If there is no predator, the prey population will grow in Smith mode. The Smith model mainly describes the growth rates of the population under the condition of limited food. The relative growth rate of population size at time is proportional to the amount of food left before the time . The food consumed by the population is mainly used for two purposes: the food needed to maintain the organism’s own survival and the food needed for population reproduction . The denominator of functional response function contains , which reflects the interaction among individuals within the predator population. The second equation implies that the predator population grow in Leslie mode and the environmental capacity is proportional to prey population density.

Developing and utilizing biological populations in a proper way is an important issue facing humanity, which can make biological resources work for human beings and ensure the sustainability of ecosystems [6–12]. People’s harvesting behavior is often not continuous, which may be once or several times at a certain time. For example, in fishery production, fishermen harvest fishes in few months and in agricultural production farmers spray pesticides at regular intervals to harvest (kill) pests. This harvesting method makes the population quantity or density be changed drastically in a short period of time, so it is more suitable to describe with impulsive differential equations. In the past 20 years, impulsive differential equations have been widely used in various biological models, for example, Zhao and Tang [13], Liu et al. [14], Li et al. [15], and Zhang et al. [16] introduced impulsive effects into the ordinary differential equation model. Li and Meng [17] and Qi et al. [18] established a class of impulsive stochastic differential equation models based on ordinary differential equations. Du and Feng [19] and Chen and Du [20] cooperated the impulsive control to the neutral predator-prey model, by using the coincidence degree theory, and they obtained the conditions for the existence of periodic solution. Then, based on [19, 20] and assuming that some parameters in system (5) are changed periodically and the population is pulse controlled, we will obtain the following neutral delay system with pulse effects:with initial conditionswhere , , is the pulse harvesting time, and is the pulse harvesting ratio. This implies that the harvest yield is in proportion to the biomass at that time. are continuous nonnegative T-periodic functions. is positive constant and .

On the contrary, biological populations multiply and thrive in nature, which will inevitably be affected by various environmental noise, so it is important to consider the effects of random disturbances on population dynamics [21–29]. In [30], the asymptotic stability of a stochastic May mutualism system was studied. In [31], Markov semigroup was used to study a stochastic ecological model of plants with infectious diseases. Authors in [32–34] focused on the long-term dynamic behavior of ecosystems affected by environmental noise. In particular, conditions for the existence of stationary distribution were obtained. Zhao et al. [35] presented an algal population growth model, where the authors not only gave the threshold conditions of permanence and extinction but also discussed the influence of environmental noise on the periodic blooms. Ignoring the influence of time delays and considering the disturbance of white noise, we construct the following stochastic Leslie predator-prey model with Smith growth:where and are independent standard Brownian motions. denote the intensity of white noise. , and are all positive and the biological significances are the same as that in model (5).

To study the existence of periodic solutions for ordinary differential equations and delay differential equations, Gaines and Mawhin’s continuation theorem based on coincidence degree theory [36] is an important tool. However, it is seldom applied in the study of the existence of positive periodic solutions of neutral delay system with pulse effects. It is generally considered difficult to prove that the nonlinear operator is compact on set and the solutions of the equation have prior bound. In this work, by using analytical methods such as the mean value theorem of integral and inequality techniques, we obtain the sufficient conditions for the existence of positive periodic solutions of system (6). In addition, considering the influence of environmental disturbance on the population, we include the white noise into the Leslie predator-prey system, establish a new stochastic Smith growth model (8), and discuss the conditions for the persistence and extinction of the system.

2. Preliminaries

First, we convert the impulse system (6) into a nonimpulsive form. In system (6), we assume that are fixed points and are real sequences such that and are T-periodic functions

Assuming and , we convert system (6) to the following system:with initial conditionswhere

Lemma 1. Suppose that and hold, then(1)If is a solution of (9), then is a solution of (6)(2)If is a solution of (6), then is a solution of (9)

Proof. (1) Suppose is a solution of system (9). Next, we prove that is the solution of system (6). When , we obtainWhen , one hasTherefore, is the solution of system (6).
(2) Similar to the proof of case (1), we could show that if is a solution of (6), then is the solution of system (9). The proof of Lemma 1 is completed.
For convenience, throughout this paper we will use the symbols:where is a periodic continuous function with period T.

3. Existence of Positive Periodic Solution of System (6)

Before discussing the existence of periodic solution of system (6), we introduce the coincidence degree theory firstly.

Let and be two Banach spaces, : Dom is a linear mapping and is a continuous mapping. If dim Ker = codim Im and Im is closed, then we call the operator is a Fredholm operator with index zero. If is a Fredholm operator with index zero and there exist continuous projections and such that and ; then, has an inverse function, and we set it as . Assume that is any open set, if is bounded and is relative compact, and then we say is -compact on .

Lemma 2 (see [36]). Let :Dom be a Fredholm mapping with index zero and is a -compact on . Furthermore, assume that(i), where , Dom(ii), for each (iii)deg, where is an isomorphismThen, the operator equation has at least one solution in Dom.

Lemma 3 (see [37]). If with and for , then function has a unique inverse satisfying with for .
We are now in a position to state and prove our main result.

Theorem 1. Assume that the following conditions hold: , for any for any , where , , where are defined in the proof The algebraic equations have finite solutions. Then, system (6) has at least one T-periodic solution.

In the following proof, we first transform (9) into equivalent system. Then, we construct the mappings and open set on Banach space and prove that the mappings satisfy Lemma 2 on . Therefore, we obtain Theorem 1.

Proof. (1) Transform (9) into an equivalent system.
Let . Then, we can translate system (9) intowhere all coefficient functions are defined in system (9). It is easy to see that if system (16) has one T-periodic solution , then is a positive T-periodic solution of system (9). So, system (6) has a T-periodic solution . Therefore, we just have to prove system (16)s has one T-periodic solution.
(2) Construct Banach space and define mappings.
Take and denote . Then, and are both Banach spaces with the norm and , respectively. Define operators , , , and in the following form, respectively, , ; , for ; , for ; andThen, Ker, Im is closed in , dimKer codimIm. and are continuous projectors such that Im Ker, . Therefore, the Fredholm mapping has a unique inverse. The generalized inverse(to ):Im KerDom is given by the following form:Then, and readwhereDistinctly, it is easy to know that and are both continuous by the Lebesgue theorem. And by using Arzela–Ascoli theorem, we know that is compact for any open bounded set . Hence, is -compact on .
(3) Construct an open set .
For the sake of using Lemma 2, we need to look for an appropriate open-bounded subset . First, we prove that the periodic solution of equation is bounded. Corresponding to the operator equation for , we haveSuppose that is a solution of (21). Now we prove that , is a constant.

Step 1. Prove that and .
Integrating (21) over the interval , we obtainandIn consideration of and , we obtainwhich together with (22) givesIn view of Lemma 3 and , we obtainwhere . From (25), we obtainwhich impliesTherefore,where . According to (25), we obtaintogether with , which implieswhere .
According to (29) and (31), we can get that there exists such thatBy means of (25) and (29), we haveOn the basis of the mean value theorem of integral calculus, we know that there exists , such thatSince , we havewhere .
Through the first equation of (21) and (25), we obtainTherefore,So, we obtainFrom (36), we knowConsequently,Through the transpose, we can obtainTogether with (32), we obtainThrough the first equation of (21) and (38), we obtainso we obtainUtilizing , we have

Step 2. Prove that and .
By means of (23) and (38), we obtainwhere , which implies thatAccording to (23) and (42), we obtainThus,From (47) and (49), there exists such thatFrom (21) and (23), it is easy to see thatThen,Through the second equation of (21), (42), and (52), we haveFrom (42), (45), (52), and (53), we obtainAt this point, we have proved that the periodic solution of equation for is bounded.
In the following part, we will prove that the constant solution of equation is bounded. According to condition , the algebraic equation sethas finite positive solutions . Therefore, for Ker, the operator equation has finite positive solutions . Suppose that , then . Therefore, the solution of equation for Ker is bounded.
Let , and take . It is easy to see that verifies condition in Lemma 2. When Ker, is a constant vector in with , we obtainThus, condition of Lemma 2 holds.
(4) Calculate the degree of coincidence.
For each , define isomorphism :Im ker so that . By simply computing, we get the following Jacobian matrix:Then, we havewhich impliesBy now, we have verified that satisfies all the requirements in Lemma 2. Thus, system (16) has at least one T-periodic solution, which implies that system (6) has at least one positive T-periodic solution. This completes the proof.

Remark 1. If the delay both are positive constants, that is to say, condition holds. At this time, the delay has no effect on the existence of periodic solutions.

Remark 2. If the neutral coefficient , that is, , at this time, condition holds. When , and hold, then the system has at least one positive periodic solution.
Next, we will study the predator-prey model with random disturbances.

4. Persistence in Mean and Extinction of System (8)

In this section, we discuss the properties of stochastic system (8) with Smith growth, the sufficient conditions for persistence and extinction are obtained.

Lemma 4 (see [38]). Let be a solution of system (8) with initial value , then .

Theorem 2. Let be a solution of system (8) with initial value . If(1), then system (8) is persistent in mean(2), then system (8) goes to extinction almost surely

Proof. (1) From the first equation of system (8) and using It’s formula, we obtainIntegrating from 0 to on both sides, we can obtainThen,where . Dividing on both sides, we have thatAccording to Lemma 4, we obtainFrom the second equation of system (8) and using It’s formula, we can obtainIntegrating from 0 to on both sides, we obtainThus,Then, we can obtainBecause is persistent, . We complete the proof of (1).
(2) By (61) and (66), we can obtainTherefore,So, system (8) goes to extinction almost surely.
Theorem 2 shows that if the intensity of environmental noise is small, the prey and predator will survive for a long time, and vice versa, and they will become extinct. Therefore, environmental noise is not conducive to the survival of the population.

5. Conclusion and Numerical Simulation

This paper mainly solves the following problems:(1)For system (6) with neutral time delay and impulse control, first we transform it into nonpulse equivalent system by transformation: . Then, by using the coincidence degree theory, the sufficient conditions for the existence of periodic solution of the system is obtained.(2)For the stochastic perturbation system (8) with Smith growth, the persistence and extinction of the system are discussed.

In the following part, we introduce numerical simulations to illustrate our main theoretical results. Firstly, we verify the existence of positive periodic solution of the neutral delay Leslie predator-prey model (16).

Example 1. (1)In system (16), we choose By simple calculation, we know that conditions and hold, and , . Obviously holds. Moreover, the algebraic equations in have only one positive solution, which together with Theorem 1 yields that Example 1 has at least one periodic solution (see Figure 1).(2)We choose in system (16), and other parameters are the same as those in Figure 1. In this case, condition is true because and are constants. Thus, there is at least one periodic solution as long as condition are true (see Figure 2). This shows that the existence of periodic solutions is not affected by the constant delays.(3)We choose in system (16), and other parameters are the same as those in Figure 1.Because the neutral coefficient , condition must be true. The system has at least one periodic solution as long as other conditions are true. In this case, the prey population grows in logistic mode, which indicates that there is at least one periodic solution in general impulsive system (see Figure 3).

(a)

(b)

(c)

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(b)

(c)

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(b)

(c)

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(b)

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(b)

Example 2. Next we conduct simulations about the persistence and extinction of the stochastic Smith growth model (8). Obviously, when , , the deterministic model corresponding to model (8) has a unique locally asymptotically stable positive equilibrium . In system (8), we choose(1)Let , , then we can know that . Both stochastic system and deterministic system exist for a long time. Thus, conclusion (1) of Theorem 2 holds (see Figure 4).(2)Let . By simple calculation, we can know that , , and then the deterministic system is persistent. However, , and then the random system is extinct. Thus, conclusion (2) of Theorem 2 holds. This shows that environmental noise is harmful to the long-term survival of the population (see Figure 5).

Data Availability

Data sharing is not applicable to this article as all datasets are hypothetical during the current study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11371230) and Shandong Provincial Natural Science Foundation of China (no. ZR2019MA003).

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Copyright © 2020 Tongtong Li and Wencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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