On the validation, stability and control of certain biological systems (original) (raw)
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Nonlinear Analysis: Real World Applications, 2010
In this paper, we consider an integrated pest management model with disease in the pest and a stage structure for its natural predator, which is subject to impulsive and periodic controls. A nonlinear incidence rate expressed in an abstract form, is used to describe the propagation of the disease, which is spread through the periodic release of infective pests, the functional response of the mature predator also being given in an abstract, unspecified form. Sufficient conditions for the local and global stability of the susceptible pest-eradication periodic solution are found by means of Floquet theory and comparison methods, the permanence of the system also being discussed. These stability conditions are shown to be biologically significant, being reformulated as balance conditions for the susceptible pest class.
On the impulsive controllability and bifurcation of a predator–pest model of IPM
Biosystems, 2008
From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator-prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington-DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.
Integrated pest management models and their dynamical behaviour
Bulletin of Mathematical Biology, 2005
Two impulsive models of integrated pest management (IPM) strategies are proposed, one with fixed intervention times and the other with these unfixed. The first model allows natural enemies to survive but under some conditions may lead to extinction of the pest. We use a simple prey-dependent consumption model with fixed impulsive effects and show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than certain critical values. The effects of pest resistance to pesticides are also studied. The second model is constructed in the light of IPM practice such that when the pest population reaches the economic injury level (EIL), a combination of biological, cultural, and chemical tactics that reduce pests to tolerable levels is invoked. Using analytical methods, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given Economic Threshold (ET). The complete expression for this periodic solution is given and the ET is evaluated for given parameters. We also show that in some cases control costs can be reduced by replacing IPM interventions at unfixed times with periodic interventions. Further, we show that small perturbations of the system do not affect the existence and stability of the periodic solution. Thus, we provide the first demonstration using mathematical models that an IPM strategy is more effective than classical control methods.
Journal of The Franklin Institute-engineering and Applied Mathematics, 2016
An Integrated Pest Management (IPM) strategy with a combination of biological and chemical tactics is more useful to reduce prevalence of pests up to a level of tolerance with minimal cost to the farmers and least loss of the environment. Therefore, in this paper, the biological and chemical control methods are impulsively used together at two different times in a two-prey one-predator SI model, where the first prey is a targeted pest and the second prey is a non-targeted neutral pest with a competitive effect to the targeted pest and serves as a food for the natural enemy. Here, the disease concerned with the targeted pest population and the incidence rate of infection is of Holling type II. Moreover, prey-dependent consumption is taken into account with impulsive releasing of infected pest and natural enemies at one time and releasing of chemicals at another time. Further, in the analysis, it was obtained that the system has a susceptible and non-target pests-eradication periodic solution which is established locally as well as globally asymptotically stable using Floquet's theorem, small-amplitude perturbation skills and comparison techniques for impulsive differential equations under some sufficient conditions, which show that susceptible and non-target pest extinct when impulse period is less than some critical value (T n) or when pulse releasing amount of infected pest and/or natural enemies is greater than their respective thresholds. We also oblige sufficient condition for the permanence of the system which implies that the species coexists when impulse period greater than the critical value (T n) or pulse releasing amount of infected pest and/or natural enemies is less than their respective thresholds. Further, the effect of growth of non-targeted pest on the dynamics of the system is studied which shows that the complex dynamics follows a periodic halfing cascade to chaos. These results provide some reliable theoretical tactics for pest management and finally are verified by performing numerical simulations using some realistic data.
Pest control through viral disease: Mathematical modeling and analysis
Journal of Theoretical Biology, 2006
This paper deals with the mathematical modeling of pest management under viral infection (i.e. using viral pesticide) and analysis of its essential mathematical features. As the viral infection induces host lysis which releases more virus into the environment, on the average 'k' viruses per host, k 2 ð1; 1Þ, the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. We prove that there exists a threshold value k 0 beyond which the endemic equilibrium bifurcates from the free disease one. Still for increasing k values, the endemic equilibrium bifurcates towards a periodic solution. We further analyse the orbital stability of the periodic orbits arising from bifurcation by applying Poor's condition. A concluding discussion with numerical simulation of the model is then presented. r
The Role of Viral Infection in Pest Control: A Mathematical Study
Bulletin of Mathematical Biology, 2007
In this paper, we propose a mathematical model of viral infection in pest control. As the viral infection induces host lysis which releases more virus into the environment, on the average 'κ' viruses per host, κ ∈ (1, ∞), so the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. There exists a threshold value κ 0 beyond which the infection persists in the system. Still for increasing the value of κ, the endemic equilibrium bifurcates towards a periodic solution, which essentially indicates that the viral pesticide has a density-dependent 'numerical response' component to its action. Investigation also includes the dependence of the process on predation of natural enemy into the system. A concluding discussion with numerical simulation of the model is also presented.
The impulsive control of a two-patch integrated pest management model
Proceedings of 6-th Edition of …, 2009
We hereby consider a two-patch SI integrated pest management model with dispersal of susceptible pests between patches, which is subject to periodic impulsive biological and chemical controls. The biological control consists in the periodic release of infective pests, in a constant amount, while the chemical control consists in periodic pesticide spraying, which causes the removal of fixed proportions of the infective and susceptible pest populations, respectively. The spread of the disease inflicted by the release of infective pests is characterized by a nonlinear incidence rate of infection expressed in an abstract, unspecified form. A sufficient condition for the local stability of the susceptible pest-eradication periodic solution is obtained through the use of Floquet theory for impulsive and periodic ordinary differential equations, the effect of population dispersal between patches upon the stability of this solution being then investigated for several particular cases.
Modeling of Insect-Pathogen Dynamics with Biological Control
Mathematical Biology and Bioinformatics
In this work, a model has been proposed to analyze the effect of wild plant species on biologically-based technologies for pest control. It is assumed that the pest species have a second food source (wild host plants) except crops. Analytical results prove that the model is well-posed as the system variables are positive and uniformly bounded. The permanence of the system has been verified. Equilibrium points and corresponding stability analysis have also been performed. Numerical figures have supported the fact that the interior steady state if it exists, remains stable for any transmission rate. Henceforth biological control has a stabilizing effect. Furthermore, the results prove that biological control is beneficial not only for wild plants but for crops too.
Bifurcation analysis of a model for biological control
Mathematical and Computer Modelling, 2008
In this paper we study the Lyapunov stability and Hopf bifurcation in a biological system which models the biological control of parasites of orange plantations. This model -an elaboration of Lotka-Volterra equations, taking into account the stages or compartments in the biological populations -was proposed by Yang and Ternes [1,2] and Ternes [3] for a study of the biological control 1 of orange plantation leaf parasites P, which are in a pre-adult stage for M, by their natural enemies L, which are in an early stage for G.
Theoretical Population Biology, 2008
Host-parasitoid models including integrated pest management (IPM) interventions with impulsive effects at both fixed and unfixed times were analyzed with regard to host-eradication, host-parasitoid persistence and host-outbreak solutions. The host-eradication periodic solution with fixed moments is globally stable if the host's intrinsic growth rate is less than the summation of the mean host-killing rate and the mean parasitization rate during the impulsive period. Solutions for all three categories can coexist, with switch-like transitions among their attractors showing that varying dosages and frequencies of insecticide applications and the numbers of parasitoids released are crucial. Periodic solutions also exist for models with unfixed moments for which the maximum amplitude of the host is less than the economic threshold. The dosages and frequencies of IPM interventions for these solutions are much reduced in comparison with the pest-eradication periodic solution. Our results, which are robust to inclusion of stochastic effects and with a wide range of parameter values, confirm that IPM is more effective than any single control tactic.