1,009 research outputs found
Robust permanence for interacting structured populations
The dynamics of interacting structured populations can be modeled by
where , , and
are matrices with non-negative off-diagonal entries. These models are
permanent if there exists a positive global attractor and are robustly
permanent if they remain permanent following perturbations of .
Necessary and sufficient conditions for robust permanence are derived using
dominant Lyapunov exponents of the with respect to
invariant measures . The necessary condition requires for all ergodic measures with support in the boundary of the
non-negative cone. The sufficient condition requires that the boundary admits a
Morse decomposition such that for all invariant
measures supported by a component of the Morse decomposition. When the
Morse components are Axiom A, uniquely ergodic, or support all but one
population, the necessary and sufficient conditions are equivalent.
Applications to spatial ecology, epidemiology, and gene networks are given
Asymptotic behavior of age-structured and delayed Lotka-Volterra models
In this work we investigate some asymptotic properties of an age-structured
Lotka-Volterra model, where a specific choice of the functional parameters
allows us to formulate it as a delayed problem, for which we prove the
existence of a unique coexistence equilibrium and characterize the existence of
a periodic solution. We also exhibit a Lyapunov functional that enables us to
reduce the attractive set to either the nontrivial equilibrium or to a periodic
solution. We then prove the asymptotic stability of the nontrivial equilibrium
where, depending on the existence of the periodic trajectory, we make explicit
the basin of attraction of the equilibrium. Finally, we prove that these
results can be extended to the initial PDE problem.Comment: 29 page
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Robust permanence for ecological equations with internal and external feedbacks.
Species experience both internal feedbacks with endogenous factors such as trait evolution and external feedbacks with exogenous factors such as weather. These feedbacks can play an important role in determining whether populations persist or communities of species coexist. To provide a general mathematical framework for studying these effects, we develop a theorem for coexistence for ecological models accounting for internal and external feedbacks. Specifically, we use average Lyapunov functions and Morse decompositions to develop sufficient and necessary conditions for robust permanence, a form of coexistence robust to large perturbations of the population densities and small structural perturbations of the models. We illustrate how our results can be applied to verify permanence in non-autonomous models, structured population models, including those with frequency-dependent feedbacks, and models of eco-evolutionary dynamics. In these applications, we discuss how our results relate to previous results for models with particular types of feedbacks
Permanence and Stability of an Age-Structured Prey-Predator System with Delays
An age-structured prey-predator model with delays is proposed and analyzed. Mathematical analyses of the model equations with regard to boundedness of solutions, permanence, and stability are analyzed. By using the persistence theory for infinite-dimensional systems, the sufficient conditions for the permanence of the system are obtained. By constructing suitable Lyapunov functions and using an iterative technique, sufficient conditions are also obtained for the global asymptotic stability of the positive equilibrium of the model
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
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