The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management

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Abstract

According to the fact integrated pest management, a Lotka–Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. We analyze such system from two cases: general case (taking IPM strategy) and special case (only choosing pesticides). In the first case, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the period of impulsive effect is less than some critical value. The condition for the permanence is also given. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. In the second case, the system by only choosing chemical pesticides, we give the conditions of existence and stability of predator-free periodic solution. We obtain that the conditions for the permanence or extinction in the system we considered are quite different from the corresponding system without impulse. Finally, we compare validity of the IPM strategy with the classical methods (only biological control or chemical control), and conclude that IPM strategy is more effective than the classical one.

Introduction

A wide range of pest control strategies is available to farmers. Integrated pest management (IPM) is a suitable approach to managing pests by combining biological, cultural, physical and chemical tools in a way that minimize economic, health and environmental risks.

Biological control [3], [6], [7], [12], [19] is the reduction in pest populations from the actions of other living organisms, often called natural enemies or beneficial species. Virtually all insect and mite pests have some natural enemies. Biological control is most effective when used with compatible pest control practices in an IPM program. One approach to biological control is to release beneficial natural enemies to control insect and mite pest. This approach is known as augmentation. In some pest situations it is a highly efficacious, cost effective and environmentally sound approach to pest management. There are two general approach to augmentation: inundative releases and inoculative release. Inundation involves releasing large numbers of natural enemies for immediate reduction of a damaging or near-damaging pest population. Inoculation involves releasing small numbers of natural enemies at intervals, sometimes throughout the period of pest activity, starting when the pest population is low. The expected outcome of inoculative release is to keep pest numbers low, never allowing them to approach an economic injury level.

The use of chemical pesticides often forms part of an integrated pest management strategy. Pesticides are useful because they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss. The key is to use pesticides in a way complements rather than hinders other elements in the strategy and which also limits negative environmental effects. It is important to understand the life cycle of a pest so that the pesticide can be applied when the pest is at its most vulnerable—the aim is to achieve maximum effect at minimum levels of pesticide.

Wherever possible, different pest control techniques should work together rather than against each other. IPM which has been proved by experiments (e.g. Van Lenteren 1995 [20]; IPM for Walnuts [18]; IPM for Alfafa hay [17]) is more effective than classical one (such as biological control or chemical control). The main purpose of this paper is to construct a simple mathematical models according to the fact of IPM and investigates the dynamics of this system. We suggest an impulsive system (see [2], [10]) to model the process of periodic releasing natural enemies and spraying pesticides in Section 2. Impulsive equations are found in almost every domain of applied science. Some impulsive equations have been recently introduce in population dynamics in relation to: vaccination [1], [4], [5], [14], [15], chemotherapeutic treatment of disease [8], [9], [11], birth pulse [13], [16]. In Section 3, we analyze the dynamical behaviors of our system. At first, we consider the system concerning IPM strategy. We show that there exists a globally asymptotically stable pest-eradication periodic solution when the period of impulsive effect is less than some threshold. The condition for the permanence is also given. By using bifurcation theory, we show the existence and stability of positive periodic solution if the pest-eradication lost its stability. Next, we consider the special case, the system by only choosing chemical pesticides. We give the conditions of existence and stability of predator-free periodic solution, permanence of the system and existence and stability of positive periodic solution. Finally, we compare validity of the IPM strategy with the classical methods (only biological control or chemical control). We conclude that IPM strategy is more effective than the classical one. A brief discussion of our results are given in the last section.

Section snippets

Model formulation

The basic model we considered is based on the following Lotka–Volterra predator–prey systemdx1(t)dt=x1(t)(r-ax1(t)-bx2(t)),dx2(t)dt=x2(t)(-d+cx1(t)),where x1(t) and x2(t) are the prey (pest) and the predator (natural enemy) population (or density), respectively, r>0 is the intrinsic growth rate of pest, a>0 is the coefficient of intraspecific competition, b>0 is the per-capita rate of predation of the predator, d>0 is the death rate of predator, c>0 denotes the product of the per-capita rate of

Global qualitative analysis for model (2.2)

Before our main result, we will give some lemmas which will be useful for our main results.

The solution of the system (2.2), denoted by x(t)=(x1(t),x2(t))T, is a piecewise continuous function x:R+R+2, x(t) is continuous on (nT,(n+1)T],nZ+ and x(nT+)=limtnT+x(t) exists. Obviously the global existence and uniqueness of solutions of system (2.2) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-hand side of system (2.2) (see Lakshmikantham, [10]).

Discussion

In this paper, according to the fact IPM, a Lotka–Volterra predator–prey model with impulsive effect at fixed moment is proposed and investigated. We analyze such system from two cases: general case (taking IPM strategy) and special case (only choosing pesticides). We have shown that the IPM strategy is more effective than the classical one.

We note that the conditions for the permanence or extinction in system (2.2) are quite different from the corresponding system without impulse. For example,

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This work is supported by National Natural Science Foundation of China(10171106).

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