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We analyze a population model for two age-structured species allowing for inter- and intra-specific competition at immature life stages. The dynamics is governed by a system of Delay Differential Equations (DDEs) recently introduced by Gourley and Liu. The analysis of this model presents serious difficulties because the right-hand sides of the DDEs depend on the solutions of a system of nonlinear ODEs, and generally cannot be solved explicitly. Using the notion of strong attractor, we reduce the study of the attracting properties of the equilibria of the DDEs to the analysis of a related two-dimensional discrete system. Then, we combine some tools for monotone planar maps and planar competing Lotka–Volterra systems to describe the dynamics of the model with three different birth rate functions. We give easily verifiable conditions for global extinction of one or the two species, and for global convergence of the positive solutions to a coexistence state.

We investigate prediction based schemes to stabilize periodic solutions to potentially chaotic systems of periodic difference equations using pulses at times being a multiple of the period. By introducing the concept of a stability domain , we obtain precise information on the possibility to stabilize given solutions, to avoid destabilizing bifurcations, as well as on the magnitude of the required control.

The timing of harvesting is a key instrument in managing and exploiting biological populations and renewable resources. Yet, there is little theory on harvest timing, and even less is known about the impact of different harvest times on the stability of population dynamics, even though this may drive population variability and risk of extinction. Here, we employ the framework proposed by Seno to study how harvesting at specific moments in the reproductive season affects not only population size but also stability. For populations with overcompensation, intermediate harvest times tend to be stabilizing (by simplifying dynamics in the case of unimodal maps and by preventing bubbling in the case of bimodal maps). For populations with a strong Allee effect, however, intermediate harvest times can have a twofold effect. On the one hand, they facilitate population persistence (if harvesting effort is low). On the other hand, they provoke population extinction (if harvesting effort is high). Early harvesting, currently considered common sense to take advantage of compensatory effects, may cut into the breeding stock when the population has not yet surpassed the critical Allee threshold. The results in this paper highlight, for the first time, the crucial interplay between harvest timing and Allee effects. Moreover, they demonstrate that harvesting with the same effort but at different moments in time can dramatically alter the impact on the population.

For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients.

A B S T R A C T Population control in some form of harvesting might be expected to reduce population size, but quite the opposite can happen due to the hydra effect. This phenomenon describes an increase in population size with increased mortality. One mechanism causing hydra effects is the temporal separation of (i) harvesting and (ii) density-dependent reproduction. Here we consider discrete-time models of these two processes. It is commonly believed that harvesting needs to precede reproduction for a hydra effect to occur. We show that, by contrast, hydra effects also take place for harvest after reproduction. Due to the timing of population census, however, the hydra effect will not be measured and thus remains ‘hidden’. As a consequence, managers may miss out on the opportunity to increase both the yield and the remaining stock of renewable resources. If harvesting aims at controlling pest species, management interventions may backfire in the sense that the pest increases rather than decreases—and, to make things even worse, this may actually go unnoticed. To remedy these undesirable consequences, we propose a modelling framework that can reveal hidden hydra effects. Our results are based on rigorous mathematical proofs that the order of two events does not matter for standard harvesting/hunting strategies.

We analyze the stabilization of an unstable periodic orbit (UPO) by periodic prediction-based control (PBC). We rigorously prove that, for 2-periodic orbits, a pulse strategy reduces the necessary control strength to stabilize the UPO. Moreover, we find that in some cases the periodic control prevents some undesirable effects induced by the PBC method. In this way, we provide an example of a dynamic Parrondoʼs paradox: the switching between two undesirable dynamics results in a nicely periodic dynamic behavior.

Preprint aceite para publicação no Journal of Theoretical Biology., The purpose of this paper is to analyze the effect of constant effort harvesting upon global dynamics of a discrete-time population model with juvenile and adult stages. We consider different scenarios, including adult-only mortality, juvenile-only mortality, and equal mortality of juveniles and adults. In addition to analytical study of equilibria of the system, we analyze global dynamics by means of an automated set-oriented rigorous numerical method. We obtain a comprehensive overview of the dynamics as the harvest rate and survival probability change. In particular, we determine the range of parameters for which the population abundance gets larger in spite of an increase in the harvest rate (so-called hydra effect), and for which subsequent increases in harvesting effort can magnify fluctuations in population abundance (destabilize it) and then stabilize it again (so-called bubble effect)., The work of both authors was partially supported by the Spanish Ministry of Science and Innovation and FEDER, Grant MTM2010-14837. Moreover, the work of P. Pilarczyk was additionally supported in part from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE-Programa Operacional Factores de Competitividade (POFC) and from the Portuguese National Funds through Fundacao para a Ciencia e a Tecnologia (FCT) in the framework of the research projects FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008) and Est-C/MAT/U10013/2011, as well as from the Research Center of Mathematics of the University of Minho through the FCT Pluriannual Funding Program.The authors thank K. Mischaikow's lab at Rutgers University (supported in part by NSF, DoE, DARPA, and AFOSR) and the Computer Science and Technology Center at Departamento de Informatica of the University of Minho for providing their cluster computing facilities.

We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright’s equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems.

We study the control of chaos in one-dimensional discrete maps as they often occur in modelling population dynamics. For managing the population, we seek to suppress any possible chaotic behavior, leading the system to a stable equilibrium. In this Letter, we make a rigorous analysis of the proportional feedback method under certain conditions fulfilled by a wide family of maps. We show that it is possible to stabilize the chaotic dynamics towards a globally stable positive equilibrium, that can be chosen among a broad range of possible values. In particular, the size of the population can be enhanced by control in form of population reduction. This paradoxical phenomenon is known as the hydra effect, and it has important implications in the design of strategies in such areas as fishing, pest management, and conservation biology.

We provide further insight on the dynamics of Clark's delayed recruitment equation, depending on the relevant parameters involved in the model. We pay special attention to the stability and bifurcations from the positive equilibrium, and to the existence and attraction properties of nontrivial cycles. A detailed analysis is worked out for a three-dimensional example.

We address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker's and Pielou's equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model.

We address the problem of existence of periodic solutions for the differential delay equationεx˙(t)+x(t)=f(x(t-1)),00,where |m|0,B>0. We show that when the map x↦f(x) has an attracting 2-cycle then the delay differential equation has a periodic solution, which is close to the square wave corresponding to the limit (as ε→0+) difference equation x(t)=f(x(t-1)).

In this paper we derive new discrete Halanay-type inequalities and show some applications in the investigation of the asymptotic behavior of nonlinear di4erence equations. In particular, di4erence equations involving the “maximum” functional are considered. ? 2003 Elsevier Ltd. All rights reserved.

In this note, we prove a discrete analogue of the continuous Halanay inequality and apply it to derive sufficient conditions for the global asymptotic stability of the equilibrium of certain generalized difference equations. The relation with some numerical schemes for functional delay differential equations is discussed.

We study a periodic boundary value problem for a first-order differential equation from a new point of view, which permits us to obtain an existence result involving upper and lower solutions with “jump” discontinuities. We present some applications of our result with illustrative examples.