Chaos to Permanence-Through Control Theory

Work by Cushing et al. \cite{Cushing} and Kot et al. \cite{Kot} demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species \cite{EVG}. We utilize present chaotic behavior and a control algorithm based on \cite{Vincent97,Vincent2001} to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from \cite{Harvesting}, a ratio-dependent one-prey, two-predator model from \cite{EVG} and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model \cite{Upad} and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

A Modified Holling-Tanner Model in Stochastic Environment

Recently, a modified version of the so called Holling-Tannermodel isintroduced in the ecological literature. A detailed account of the deterministic dynamicsof this model is presented. The growth rates of the prey and predator are then perturbedby Gaussian white noises to take into account the effect of fluctuating environment. Theresulting stochastic model is cultured by the technique of statistical linearization andcriteria for non-equilibrium fluctuation and stability arederived. Numerical simulationsare carried out. The implications of our analytical findingsare addressed critically

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Stability and Bifurcation Analysis of a Delayed Three Species Food Chain Model with Crowley-Martin Response Function

In this paper we have studied the dynamical behaviors of three species prey-predator system. The interaction between prey and middle-predator is Crowley-Martin type functional response. Positivity and boundedness of the system are discussed. Stability analysis of the equilibrium points is presented. Permanence and Hopf-bifurcation of the system are analyzed under some conditions. The effect of discrete time-delay is studied, where the delay may be regarded as the gestation period of the super-predator. The direction and the stability criteria of the bifurcating periodic solutions are determined with the help of the normal form theory and the center manifold theorem. Extensive numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically

Competitive response in interference competition models

Este trabajo está enmarcado en la ecología matemática. En particular, en los modelos de competencia entre especies, que es una de las tres grandes formas de interacción entre especies. El punto de partida es el modelo clásico de Lotka y Volterra que asume implícitamente que 1. La tasa de crecimiento per cápita disminuye linealmente con el incremento del tamaño de cualquiera de las poblaciones que compiten. 2. Las interacciones entre especies son instantáneas, en el sentido de ser independientes del tamaño de la población con que se compite. 3. Los individuos están bien mezclados y cualquier individuo de una especie puede interactuar con cualquier individuo de la otra. Estas asunciones son totalmente válidas en una primera aproximación, pero distan mucho de ser universales. En esta tesis se presentan nuevas formulaciones del modelo clásico de competencia aplicables a aquellas situaciones en las que no se cumple al menos una de esas tres hipótesis. En concreto, se analizan tres modelos: 1. Un modelo en el que se tiene en cuenta que competir consume tiempo. 2. Un modelo en el que incorpora a la competencia el mecanismo de defensa grupal. 3. Un modelo de competencia para organismos sésiles (aquellos que no se desplazan). En los dos primeros modelos se introduce una respuesta competitiva que extienden el modelo clásico en el mismo sentido en que la llamada respuesta funcional opera en modelos de depredador-presa. El tercero extiende, en cierto modo, modelos publicados muy recientemente que incorporan estructura social a las interacciones entre especies no sésiles. A grandes rasgos, los resultados obtenidos extienden los resultados disponibles en tres sentidos: • Los tres modelos permiten escenarios de bi y tri-estabilidad, en los que las especies pueden o bien coexistir o bien extinguirse una de ellas en función del tamaño inicial de cada población. • En los tres modelos la presión de la competencia inter-especies es menor que en el modelo clásico. • Como consecuencia de lo anterior, la región del espacio de parámetros que permite estados de equilibrio de coexistencia es mayor que en el caso clásico. Cada modelo tiene sus especificidades que se traducen en interpretaciones ecológicas concretas

A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey

In this manuscript, the dynamics of a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey are studied. The Caputo fractional-order derivative is used as the operator of the model by considering its capability to explain the present state as the impact of all of the previous conditions. Three biological equilibrium points are successfully identified including their existing properties. The local dynamical behaviors around each equilibrium point are investigated by utilizing the Matignon condition along with the linearization process. The numerical simulations are demonstrated not only to show the local stability which confirms all of the previous analytical results but also to show the existence of periodic signal as the impact of the occurrence of Hopf bifurcation

An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies