Global stability and optimisation of a general impulsive biological control model

Dynamic analysis and control of a rice-pest system under transcritical bifurcations

A decision model is developed by adopting two control techniques, combining cultural methods and pesticides in a hybrid approach. To control the adverse effects in the long term and to be able to evaluate the extensive use of pesticides on the environment and nearby ecosystems, the novel decision model assumes the use of pesticides only in an emergency situation. We, therefore, formulate a rice-pest-control model by rigorously modelling a rice-pest system and including the decision model and control techniques. The model is then extended to become an optimal control system with an objective function that minimizes the annual losses of rice by controlling insect pest infestations and simultaneously reduce the adverse impacts of pesticides on the environment and nearby ecosystems. This rice-pest-control model is verified by analysis, obtains the necessary conditions for optimality, and confirms our main results numerically. The rice-pest system is verified by stability analysis at equilibrium points and shows transcritical bifurcations indicative of acceptable thresholds for insect pests to demonstrate the pest control strategy.

Influence of the components of propagule pressure, Allee effects, and stochasticity on the time to establish introduced populations

Forecasting whether individuals of an introduced population will succeed to establish is a challenge in invasion and conservation biology. The present work aims to decouple the impact of the components of propagule pressure on the time for population establishment in the presence of Allee effects and stochasticity in propagule sizes. The mean first passage time (MFPT) for a population to reach a viable size is used as a measure of the establishment success for the introduction processes involving periodic introductions. By fixing the introduction rate (mean number of introduced individuals per unit time) and varying the period of introduction from small ranges (small and frequent introductions) to large ones (infrequent and large releases), we study the influence of introduction distribution over time. These patterns of introduction are compared in a semi-stochastic model by observing which factors minimize the MFPT from an initially absent population, and hence, ensure the fastest population establishment. We investigate the influence on these minima of the introduction rate, variability in the introduction sizes, and occurrence of catastrophes that temporarily wipe out the population. Whereas most investigated cases show that infrequent and large introductions favor population establishment as expected, small and frequent introductions are preferred when the introduction rate is large and/or the variability in the introduction size is strong. Moreover, we observed counterintuitively that catastrophes strongly increase MFPT at small periods of introduction. In addition, we showed that stochasticity in introduction tends to increase the MFPT except when the introduction rate is small and introductions are evenly spread out in time.

Coupling epidemiological and tree growth models to control fungal diseases spread in fruit orchards

Agronomic practices can alter plant susceptibility to diseases and represent a promising alternative to the use of pesticides. Yet, they also alter crop quality and quantity so that the evaluation of their efficacy is not straightforward. Here we couple a compartmental epidemiological model for brown rot diffusion in fruit orchards with a fruit-tree growth model explicitly considering the role of agronomic practices over fruit quality. The new modelling framework permits us to evaluate, in terms of quantity and quality of the fruit production, management scenarios characterized by different levels of regulated deficit irrigation and crop load. Our results suggest that a moderate water stress in the final weeks of fruit development and a moderate fruit load provide effective control on the brown rot spreading, and eventually guarantee monetary returns similar to those that would be obtained in the absence of the disease.

AN INTEGRATED PEST MANAGEMENT MODEL WITH DOSE-RESPONSE EFFECT OF PESTICIDES

Considering the delayed response to pesticide applications and the long-term residual effects of pesticides after the deployment of a pest management strategy, this paper develops a pollutant-discharge model to simulate pesticide spraying and analyze the effect of releasing natural enemies of the pest. The following two different control strategies are discussed: (1) the frequency of spraying pesticides is higher than that of releasing natural enemies, and (2) the frequency of releasing natural enemies is higher than that of spraying pesticides. For different control strategies, the sufficient conditions of locally asymptotic stability and globally asymptotic stability of the pest-eradication periodic solution are obtained. Using numerical simulations, we analyze the sensitivity of the threshold condition with respect to the parameters, identify the major factors affecting pest control and provide guidance for decision-making in pest management. Finally, we compare the control strategies and analyze which strategy is optimal as the most significant control parameters are varying.

Theoretical study and control optimization of an integrated pest management predator-prey model with power growth rate

This work presents a pest control predator-prey model, where rate of change in prey density follows a scaling law with exponent less than one and the control is by an integrated management strategy. The aim is to investigate the change in system dynamics and determine a pest control level with minimum control price. First, the dynamics of the proposed model without control is investigated by taking the exponent as an index parameter. And then, to determine the frequency of spraying chemical pesticide and yield releases of the predator, the existence of the order-1 periodic orbit of the control system is discussed in cases. Furthermore, to ensure a certain robustness of the adopted control, i.e., for an inaccurately detected species density or a deviation, the control system could be stabilized at the order-1 periodic orbit, the stability of the order-1 periodic orbit is verified by an stability criterion for a general semi-continuous dynamical system. In addition, to minimize the total cost input in pest control, an optimization problem is formulated and the optimum pest control level is obtained. At last, the numerical simulations with a specific model are carried out to complement the theoretical results.

Biocontrol in an impulsive predator-prey model

We study a model for biological pest control (or "biocontrol") in which a pest population is controlled by a program of periodic releases of a fixed yield of predators that prey on the pest. Releases are represented as impulsive increases in the predator population. Between releases, predator-pest dynamics evolve according to a predator-prey model with some fairly general properties: the pest population grows logistically in the absence of predation; the predator functional response is either of Beddington-DeAngelis type or Holling type II; the predator per capita birth rate is bounded above by a constant multiple of the predator functional response; and the predator per capita death rate is allowed to be decreasing in the predator functional response and increasing in the predator population, though the special case in which it is constant is permitted too. We prove that, when the predator functional response is of Beddington-DeAngelis type and the predators are not sufficiently voracious, then the biocontrol program will fail to reduce the pest population below a particular economic threshold, regardless of the frequency or yield of the releases. We prove also that our model possesses a pest-eradication solution, which is both locally and globally stable provided that predators are sufficiently voracious and that releases occur sufficiently often. We establish, curiously, that the pest-eradication solution can be locally stable whilst not being globally stable, the upshot of which is that, if we delay a biocontrol response to a new pest invasion, then this can change the outcome of the response from pest eradication to pest persistence. Finally, we state a number of specific examples for our model, and, for one of these examples, we corroborate parts of our analysis by numerical simulations.

An impulsively controlled pest management model with n predator species and a common prey

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.

THE EFFECTS OF TIMING OF PULSE SPRAYING AND RELEASING PERIODS ON DYNAMICS OF GENERALIZED PREDATOR-PREY MODEL

Based on the facts of releasing natural enemies and spraying pesticides at different time points, we propose a generalized predator-prey model with impulsive interventions. The threshold values for the existence and stability of pest eradication periodic solution are provided under the assumptions of releasing natural enemies either more or less frequent than spray. In order to address how the different pulse time points, control tactics affect the pest control (i.e. the threshold value), the Holling Type II Lotka-Volterra predator-prey system, as an example, with impulsive intervention at different time points are investigated carefully. The numerical results show how the threshold values are affected by the factors including instantaneous killing rates of pesticides on pests and natural enemies, the release rate of natural enemies and release constant, timing of pesticide application and timing of release period. Furthermore, it is confirmed that the system has the coexistences of pests and natural enemies for a wide range of parameters and with quite different pest amplitudes.

Two models of interfering predators in impulsive biological control

In this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters.

A note on semi-discrete modelling in the life sciences

Semi-discrete models are a particular class of hybrid dynamical systems that undergo continuous dynamics most of the time but repeatedly experience discrete changes at some given moments. In the life sciences, since the first semi-discrete model was derived to describe population dynamics by Beverton & Holt (Beverton & Holt 1957 In Fisheries investigations, series 2, vol. 19), a large body of literature has been concerned with such modelling approaches. The aim of the present contribution is twofold. On the one hand, it provides a comprehensive introduction to semi-discrete modelling through two illustrative examples: the classical work by Beverton and Holt is recalled and an original example on immigration in a population model affected by a strong Allee effect is worked out. On the other hand, a short overview of the different applications of semi-discrete models in the life sciences is proposed.