Abstract
In this work we consider asize-structured cannibalism model with the model ingredients(fertility, growth, and mortality rate) depending on size(ranging over an infinite domain) and on ageneral function of the standing population (environmentalfeedback). Our focus is on the asymptotic behavior of thesystem, in particular on the effect of cannibalism on the long-termdynamics. To this end, we formally linearize the system aboutsteady state and establish conditions in terms of the modelingredients which yield uniform exponential stability of the governing linear semigroup.We also show how the point spectrum of the linearized semigroup generatorcan be characterized in the special case of a separable attackrate and establish a general instability result.Further spectral analysis allows us to give conditions forasynchronous exponential growth of the linear semigroup.
Highlights
Cannibalism is a phenomenon observed among many species, e.g. certain fish populations
Diekmann et al have developed a general mathematical framework to study analytical questions for structured populations, including those pertaining to linear/nonlinear stability of population equilibria. In this context it was recently proven for large classes of structured population models, formulated as integral equations, that the nonlinear stability/instability of a population equilibrium is completely determined by its linear stability/instability, a result commonly referred to as the “Principle of Linearized Stability”
In this work we have studied the asymptotic behavior of solutions of a linearized size-structured cannibalism model, recently introduced in [7]
Summary
Cannibalism is a phenomenon observed among many species, e.g. certain fish populations. Well-posedeness of structured partial differential equation models with infinite dimensional environmental feedback variables is in general an open question. Studying the linear stability of stationary solutions for models with infinite dimensional interaction variables by spectral analysis (as done previously for simpler cases in [11, 12, 20]) has proven difficult since eigenvalues are not given by an explicitly available characteristic equation (see [13]). For models with infinite dimensional interaction variables one can usually not formulate elegant necessary and sufficient conditions for the existence of steady state solutions. In this situation, one can construct positive stationary solutions by perturbation of solutions with α ≡ const. Throughout the rest of the paper we will tacitly assume that stationary solutions of the required regularity are available
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