Null controllability of a coupled model in population dynamics

Abstract

ly, a functional response can be defined as the relationship between an individual’s rate of consumption (here we talk about a consumption of predator) and food’s density (i.e., prey’s density). This amounts to saying that a functional response reflects the capture ability of the predator to prey or in other words, the functional response is introduced to describe the change in the rate of consumption of prey by predator when the density of prey varies. In the plotting point of view, each type of functional response I, II or III has a special characteristic. In fact, type I, or the linear case of the predator response, is the situation when the plot of the number of prey consumed (per unit of time) as a function of prey density shows a linear relationship between the number of prey consumed and the prey density. The Holling type II, called also concave upward response, is the case when the gradient of the curve decreases monotonically with increasing prey density, probably saturating at a constant value of prey consumption. For information, the Lotka-Volterra model involving this functional response is known as the Rosenzweig-MacArthur model. The type III response is known between the specialists of population dynamics as the sigmoid response having a concave downward part at low food density. Actually, for the Holling III, a sigmoidal behavior occurs when the gradient of the curve first increases and then decreases with increasing prey density. This behavior is due to the “learning behavior” in the predator population. Now, we address some “ecological” interpretations of the three first Holling types functional responses. The type I response is the result of simple assumption that the probability of a given predator (usually the passive one) encountering prey in a fixed time interval [0, Tt] within a fixed spatial region depends linearly on the prey density. This can be expressed under the form Y = aTsX, where Y is the amount of prey consumed by one predator, X is the prey density, Ts is the time available for searching and a is a constant of proportionality, termed as the discovery rate (which is in our case represented by the parameter b). In the absence of need to spend time handling the prey, all the time can be used for searching, i.e., Ts = Tt, and we have the type I response: assuming that the predators (having the density P ) act independently, in time Tt the total amount of prey will be reduced by quantity aTtXP . In addition, if each predator requires a handling time h for each individual prey that is consumed, the time available for searching Ts is reduced: Ts = Tt−hY . Taking into account the expression of Y in response type I, this leads to Y = aTtX − ahXY and this implies Y = aTtX/(1 + ahX) and this is exactly the type II response. Therefore, in the interval [0, Tt] the total amount of prey is reduced by the quantity aTtXP/(1 + ahX). Let us point out that the term “ah” is dimensionless and can be interpreted as the ability of a generic predator to kill and consume a generic prey and it possesses the following characteristics times: “ah” is large if the handling time h is much longer than the typical discovery time 1/a and “ah” is small in the opposite limit; in this case 8 Online first the type II response is reduced to type I. The Holling type III functional response can be viewed as a generalization of type II and takes the form aTtX /(1 + ahX) with k > 1. In literature, this response is stimulated by supposing that learning behavior occurs in the predator population with a consequent increase in the discovery rate as more encounters with prey occurs (see [25] for more details). To see to wingspan of the Lotka-Volterra models from many sides of investigations, we provide a nonexhaustive list of some works dealing with crucial questions, which are discussed widely. We begin with the system whose functional response is Holling I. One of the important problems which takes a special attention, is the study of the steady states and more accurately, in [40] a prey-predator system with nonlinear diffusion effects is considered. Such nonlinear diffusion effects have an impact on a biological species as well as their resource-biomass (i.e., the capacity of their environment). Herein, the workers assume that the dispersive force and the diffusion depend on population pressure from other species. The question of equilibrium of Lotka-Volterra systems with Holling type I functional type response takes also a broad study theoretically and numerically in [54], specially the interior one, as well as their dynamical behavior such as the cyclic-fold, saddle-fold, homoclinic saddle connection. The Holling I introduced here is from the range of the so-called Beddington-DeAngelis functional response. Remaining in the type I, the authors in [44] tried to prove the existence of an asymptotically stable pest-eradication for a prey-predator system modeled by an ordinary differential equation, when the impulsive period is some critical value less by implementing Floquet theorem and a small amplitude perturbation method. Such a solution of eradication is somehow the mixing between a synthetic strategy (insecticides or pesticides for instance) and biological control, e.g. the natural enemies “killing” the dangerous pests (the prey here) without causing a serious damages to the two population densities (see also [59] for a similar study). Even the similarity appearance between their curves, functional responses of type I and type II have two considerable differences: the first one was pointed out before and it concerns the predator time handling of prey. Contrary to the Holling type II, the time handling is missed for predators in type I, which means that the consumers have a little difficulty capturing and assimilating prey but they switch their time to other activities once their ingestion rate is great enough to satisfy their energetic needs. The second difference is in the dynamical behavior. In fact, while the Holling type II displays the local Hopf bifurcation, the Holling type I makes clear a global cyclic-fold bifurcation. These differences between Holling I and Holling II, in particular the first, lead a numerous works to take into account the predator time handling in their different models. We emphasize here that Holling II possesses a generalization, which is exactly Beddington-DeAngelis functional response cited previously. This functional takes the form ΦBD(N,P ) = cN/(e+N + h1P ), where N Online first 9 and P should be the densities of prey and predator, while e stands for the halfsaturation constant, i.e., the amount of prey at which the per capita predation rate is half of its maximum c and h1 is a positive constant (see [54] for further details about this functional response). Among the works interested in Lotka-Volterra with Holling type II we cite for instance [48], [55]. In [55], a statistical study was presented to see if one can replace Holling type II by functional response from the type of Beddington-DeAngelis, Crowly-Martin or Hassel-Valey model for a divers cases related to the predator feeding rate. Peng et al. in [48] were concerned with the question of the steady-states of some reaction-diffusion models and they established the non-existence of a non-constant equilibrium solutions of two prey-predator systems with Holling II when the interaction between the two populations is strong as they claimed and where the constant measuring the ability of generic predator to kill and consume generic prey is equal to 1. By the way, a wide classical ecological literature assumed that mathematical models with Holling type II (or in general the non-sigmoid) functional response involving a diffusion terms match thoroughly in description of the pattern formation of a phytoplankton-zooplankton system. The affirmative answers are basically related to experiments realized in laboratories on zooplankton feeding, which are carried out in small-sized containers or bottles. But if one wants to investigate zooplankton grazing control in real ecosystems (may be the oceans), it will be more relevant to introduce the Holling type III response as stated in the introduction of [46]. Actually, the main focus of [46] was to set a generic model which explains the observed alteration of type between the different functional responses of plankton systems and gives, as he presumed, an evidence that for such a system the Holling type III is more adequate than other kinds. In the vocation of well-posedness, the global existence, uniqueness and the boundedness of a strong solution of partial differential equation with a special case of Holling III was brought out in [11]. This strong solution was approximated numerically using a spectral method and a Runge-Kutta time solver. The modelling using the Holling type III does not stop here, it can play also a crucial role to model the entomophagous species (see [16] for more details). But when a functional response describes the interaction between predator and prey when the prey exhibits group defense (like buffalo) or has ability to hide itself (like chameleon), then we talk about the Holling type IV functional response or the so-called Monod-Haldane function. This function takes the form mX/(γ + b1X +X ), where X is the prey density, m > 0 is the complete saturation, whereas γ and b1 denote, respectively, the half-saturation constant and b1 the prey environmental carrying capacity. A space independent system of LotkaVolterra kind using type IV was under consideration in [45] and the principal purpose of this item is to assess the impact of the harvesting on equilibria of both prey and 10 Online first predator populations. The quandaries used here are, as the authors cited, based on the dynamical theory combined with a technique of Hopf bifurcation. A numerical analysis is provided to compare the dependence of the dynamical behavior on the harvesting effort for the prey between Holl