Measure-valued solutions to size-structured population model of prey controlled by optimally foraging predator harvester

Abstract

Radon-measure-valued solutions to a size structured population model of the McKendrick–von Foerster-type are analytically studied under general assumptions on individuals’ growth, birth and mortality rates. The model is used to describe changes in size structure of zooplankton when prey size-dependent mortality rate is a consequence of a planktivorous fish foraging in low prey-density environment (commonly found in predator-controlled populations). The model of foraging is based on the optimization of the rate of net energy intake as a function of predator speed. Mortality is defined as an operator on a metric space of nonnegative Radon measures equipped with the bounded Lipschitz distance. The solutions to the size structured model of zooplankton population are studied analytically and numerically. Numerical solutions (derived using the Escalator Boxcar Train (EBT)-like schema), in particular those starting from Dirac deltas corresponding to distinct cohorts, exhibit regularization in time and convergence to the same stationary state.

A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

“…Proof. The proof follows from similar arguments of Proposition 2.6 in [35] with making use of g(µ)(x) > 0 for all x. Indeed, since g(µ)(x) > 0 for all x we have lim t−→∞ l 0 (t) = ∞.…”

“…and repeat the proof of Theorem 3.1 verbatim to obtain that µ t is supported in hN for any t ≥ 0. It follows that µ t can be written as in (35). Equation (36) follows from (13) taking a C 1 test-function, φ, constant in time and supported in (lh − h, lh + h) such that φ(lh) = 1.…”

“…In this paper, we adopt similar assumptions on our model ingredients as in [20], but will prove well-posedness using a xed-point approach presented in [9]. Finally, for a structured model without coagulation or fragmentation, [35] proves that solutions are absolutely continuous to the left of the zero characteristic curve. Under similar assumptions, we will extend this result to structured coagulation-fragmentation equations.…”

“…This extension allows the unication of the discrete model ( 1) and the continuous (density) model ( 2) under the same framework. In recent years, the space of Radon measures equipped with the bounded Lipschitz norm has been used in the study of population dynamics [17,18,32,35]. While many population models have been studied intensely in this setting, the study of coagulation-fragmentation equations in this space is sparse.…”

“…In the recent paper [35], it was shown that the steady state solution of a size-structured population model (i.e. model (11) with K ≡ F ≡ 0) with positive model ingredients is absolutely continuous with respect to the Lebesgue measure.…”