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In the 1990s, after a series of experiments, the behavioural psychologist and economist Daniel Kahneman and his colleagues formulated the following peak-end evaluation rule: the remembered utility of pleasant or unpleasant episodes is accurately predicted by averaging the peak (most intense value) of instant utility (or disutility) recorded during an episode and the instant utility recorded near the end of the experience (Kahneman et al 1997 Q. J. Econ. 112 375–405). Based on this rule, we propose a mathematical model for the time evolution of the experienced utility function u = u ( t ) given by the scalar differential equation u ′ ( t ) = a u ( t ) + b max { u ( s ) : s ∈ [ t − h , t ] } + f ( t ) ( * ) , where f represents exogenous stimuli, h is the maximal duration of the experience, and a , b ∈ R are some averaging weights. In this work, we study equation ( ∗ ) and show that, for a range of parameters a , b , h and a periodic sine-like term f, the dynamics of ( ∗ ) can be completely described in terms of an associated one-dimensional dynamical system generated by a piece-wise continuous map from a finite interval into itself. We illustrate our approach with two representative examples. In particular, we show that the utility u(t) (e.g. ‘happiness’, interpreted as hedonic utility) can exhibit chaotic behaviour.

Proportional threshold harvesting (PTH) refers to some control rules employed in fishing policies, which specify a biomass level below which no fishing is permitted (the threshold), and a fraction ...

Threshold harvesting removes the surplus of a population above a set threshold and takes no harvest below the threshold. This harvesting strategy is known to prevent overexploitation while obtaining higher yields than other harvesting strategies. However, the harvest taken can vary over time, including seasons of no harvest at all. While this is undesirable in fisheries or other exploitation activities, it can be an attractive feature of management strategies where removal interventions are costly and desirable only occasionally. In the presence of population fluctuations, the issue of variable harvests and population sizes becomes even more notorious. Here, we investigate the impact of threshold harvesting on the dynamics of both population size and harvests, especially in the presence of population cycles. We take into account semelparous and iteroparous life cycles, Allee effects, observation uncertainty, and demographic as well as environmental stochasticity, using generic mathematical models in discrete time. Our results show that threshold harvesting enhances multiple forms of population stability, namely persistence, constancy, resilience, and dynamic stability. We discuss plausible choices of threshold values, depending on whether the aim is resource exploitation, pest control, or the stabilization of fluctuations. Agencia Estatal de Investigación | Ref. MTM2017-85054-C2-1-P

It is now recognized that the personal well-being of an individual can be evaluated numerically. The related utility (“happiness”) profile would give at each instant t the degree u(t) of happiness. The moment-based approach to the evaluation of happiness introduced by the Nobel laureate Daniel Kahneman establishes that the experienced utility of an episode can be derived from real-time measures of the pleasure and pain that the subject experienced during that episode. Since these evaluations consist of two types of utility concepts: instant utility and remembered utility, a dynamic model of happiness based on this approach must be defined by a delay differential equation. Furthermore, the application of the peak-end rule leads to a class of delay-differential equations called differential equations with maxima. We propose a dynamical model for happiness based on differential equations with maxima and provide results which shed some new light on important experimental observations. In particular, our model supports the U-shaped profile of the age-happiness curve, which is a widely observed pattern: well-being is high in youth, falls to a minimum in midlife (midlife crisis), and rises again in old age.

In an attempt to explain experimental evidence of chaotic oscillations in blood cell population, A. Lasota suggested in 1977 a discrete-time one-dimensional model for the production of blood cells, and he showed that this equation allows to model the behavior of blood cell population in many clinical cases. Our main aim in this note is to carry out a detailed study of Lasota's equation, in particular revisiting the results in the original paper and showing new interesting phenomena. The considered equation is also suitable to model the dynamics of populations with discrete reproductive seasons, adult survivorship, overcompensating density dependence, and Allee effects. In this context, our results show the rich dynamics of this type of models and point out the subtle interplay between adult survivorship rates and strength of density dependence (including Allee effects).

Threshold-based harvesting strategies tend to give high yields while protecting the exploited population. A significant drawback, however, is the possibility of harvesting moratoria with their socio-economic consequences, if the population size falls below the threshold and harvesting is not allowed anymore. Proportional threshold harvesting (PTH) is a strategy, where only a fraction of the population surplus above the threshold is harvested. It has been suggested to overcome the drawbacks of threshold-based strategies. Here, we use discrete-time single-species models and rigorously analyze the impact of PTH on population dynamics and stability. We find that the population response to PTH can be markedly different depending on the specific population model. Reducing the threshold and allowing more harvest can be destabilizing (for the Ricker and Hassell map), stabilizing (for the quadratic map), or both (for the generalized Beverton-Holt map). Similarly, management actions in the form of increasing the threshold do not always improve population stability-this can also be due to bistability. Our results therefore emphasize the importance of a rigorous analysis in investigating the impact of PTH on population stability.

We study a discrete-time model for a population subject to harvesting. A maximum annual catch H is fixed, but a minimum biomass level T must remain after harvesting. This leads to a mathematical model governed by a continuous piecewise smooth map, whose dynamics depend on two relevant parameters H and T. We combine analytical and numerical results to provide a comprehensive overview of the dynamics with special attention to discontinuity-induced (border-collision) bifurcations. We also discuss our findings in the context of harvest control rules.

A one-dimensional discrete-time dynamical system can be also seen as a recurrence, a difference equation, or an iteration scheme; and sometimes theoretical results come from different contexts. In this paper, I present a short survey about a particular family of one-dimensional maps that I have found in different situations. First, I introduce and explain the various motivations for the equation, and then I state some relevant results, with suitable references. Finally, I include some open problems and some words of caution about a series of recent poor-quality papers that, pretending to rediscover this equation, provide trivial results.

We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.

The gamma-Ricker model is one of the more flexible and general discrete-time population models. It is defined on the basis of the Ricker model, introducing an additional parameter $$\gamma >0$$ . For some values of this parameter ( $$\gamma \le 1)$$ , population is overcompensatory, and the introduction of an additional parameter gives more flexibility to fit the stock–recruitment curve to field data. For other parameter values ( $$\gamma >1$$ ), the gamma-Ricker model represents populations whose per-capita growth rate combines both negative density dependence and positive density dependence. The former can lead to overcompensation and dynamic instability, and the latter can lead to a strong Allee effect. We study the impact of the cooperation factor in the dynamics and provide rigorous conditions under which increasing the Allee effect strength stabilizes or destabilizes population dynamics, promotes or prevents population extinction, and increases or decreases population size. Our theoretical results also include new global stability criteria and a description of the possible bifurcations.

One-dimensional discrete-time population models are often used to investigate the potential effects of increasing harvesting on population dynamics, and it is well known that suitable harvesting rates can stabilize fluctuations of population abundance. However, destabilization is also a possible outcome of increasing harvesting even in simple models. We provide a rigorous approach to study when harvesting is stabilizing or destabilizing, considering proportional harvesting and constant quota harvesting, that are usual strategies for the management of exploited populations. We apply our results to some of the most popular discrete-time population models (quadratic, Ricker and Bellows maps). While the usual case is that increasing harvesting is stabilizing, we prove, somehow surprisingly, that increasing values of constant harvesting can destabilize a globally stable positive equilibrium in some cases; moreover, we give a general result which ensures that global stability can be shifted to observable chaotic dynamics by increasing one model parameter, and apply this result to some of the considered harvesting models.

The Pielou equation is a well-known discrete-time population model in which the per capita growth rate depends on the population size, but the density dependence operates with a delay of d generations. Thus, the between-year dynamics are governed by a difference equation of order $$d+1$$ . The main result in this paper establishes the global stability of the unique positive equilibrium for a generalization of the two-dimensional Pielou equation. Our proof is based on a rather natural combination of two techniques which could be, in principle, applicable to obtain global asymptotic stability in other problems: some dominance conditions and the determination of a first integral for a related equation, which turns out to be a quasi-Lyanupov function for the generalized Pielou equation. We provide additional results on the global dynamics of the generalized Pielou equation for dimensions higher than two, and discuss its relationship with other families of difference equations traditionally employed for modelling population dynamics.

Population abundance of many species is controlled by a combination of density-dependent processes during different periods of the annual cycle. In the context of population exploitation or conservation programs, sequential density dependence has the potential to dramatically change population responses to harvesting. Looking for a better understanding of the potential effects of harvesting on the dynamics of seasonal populations, we carry out a theoretical analysis of a discrete model for a semelparous population with an annual cycle involving three discrete density-dependent events: breeding, natural mortality, and harvesting. Our study reveals how the interplay between the model parameters determines the importance of harvest timing on stability and population abundance, especially when two nontrivial stable equilibria coexist. We address the possibility for compensatory mortality and report different forms of the hydra effect, including non-smooth ones due to catastrophic shifts. These drastic switches may include hysteresis, which has important implications for conservation goals. Regarding variability, we show that increasing the harvesting effort may either stabilize or destabilize the population, and these effects strongly depend on harvest timing and natural mortality rates. Our results also emphasize the importance of sampling populations after every discrete event occurs during one cycle. Indeed, though the dynamics are not affected by census timing, the model shows that changes in population abundance in response to changes in harvesting pressure are substantially different depending on when population is sampled. Thus, a manager would receive different (and sometimes contradictory) messages depending on census time, which could lead to managing mistakes.

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.

This is a post-peer-review, pre-copyedit version of an article published in Qualitative Theory of Dynamical Systems. The final authenticated version is available online at: https://doi.org/10.1007/s12346-018-00314-4 We present a new formula that makes it possible to get sharp global stability results for one-dimensional discrete-time models in an easy way. In particular, it allows to show that the local asymptotic stability of a positive equilibrium implies its global asymptotic stability for a new family of difference equations that finds many applications in population dynamics, economic models, and also in physiological processes governed by delay differential equations. The main ingredients to prove our results are the Schwarzian derivative and some dominance arguments The research of Sebastián Buedo-Fernández has been partially supported by Ministerio de Educación, Cultura y Deporte of Spain (Grant No. FPU16/04416), Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia (Grant Nos. GRC2015/004 and R2016/022), and Agencia Estatal de Investigación of Spain (Grant MTM2016-75140-P, cofunded by European Community fund FEDER) SI

Using two different discrete-time models for seasonal populations, we review the potential effects of harvesting in regard to population abundance, stability, and extinction, and we emphasize how these effects strongly depend on harvest timing. We also stress the fact that census timing is crucial, and populations should be measured as many times as a discrete event occurs during the year cycle.

The main aim of this paper is to establish sharp global stability conditions for the positive equilibrium of a well-known model of economic growth when a delay is considered in the production function. In order to deal with a broad scenario, we establish some results of global attraction for a general family of differential equations with variable delay; for it, we use the notion of strong attractor, which allows us to simplify the proofs, as well as to generalize previous results. Our study reveals that sometimes production delays are not able to destabilize the positive equilibrium, even if they are large. In other cases, the stability properties of the equilibrium depend on the interaction between the delay and other relevant model parameters, leading sometimes to stability windows in the bifurcation diagram. The research of Sebastián Buedo-Fernández has been partially supported by Ministerio de Educación, Cultura y Deporte of Spain (grant FPU16/04416), Xunta de Galicia and European Community fund FEDER (grants ED481A-2017/030, GRC2015/004, R2016/022), and AEI of Spain (grant MTM2016-75140-P). Eduardo Liz acknowledges the support of the research grant MTM2017–85054–C2–1–P (AEI/FEDER, UE) SI

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.

We show that in discrete models of Hopfield type some properties of global stability and chaotic behaviour are coded in the dynamics of a related one-dimensional equation. Using this fact, we obtain some new results on stability and chaos for a system of delayed neural networks; some relevant properties of our results are that we do not require monotonicity properties in the activation function, we allow any architecture of the network, and the conclusions are independent of the size of the time delays.

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 describe the asymptotic behaviour and the stability properties of the solutions to the nonlinear second-order difference equation for all values of the real parameters a, b, and any initial condition .

We analyze the effect of increasing mortality in usual delayed recruitment models of the form x′(t) = −δx(t) + f (x(t − τ)). We consider constant effort harvesting, and discuss the phenomenon of bubbling, which appears when for parameters δ in an interval I there exists a unique positive stationary point K(δ) in the phase space \({C([-\tau,0],\mathbb{R})}\), and there exist δ1 < δ2 in I such that K(δ) is locally stable for \({\delta\in I{\setminus}[\delta_1,\delta_2]}\), and K(δ) is unstable for \({\delta\in(\delta_1,\delta_2)}\). We give a definition of bubbling in a more abstract setting, and show that it naturally appears in a variety of equations. For some nonlinearities rigorous proofs are available for the description of bubbles, for other nonlinearities we give numerical results to indicate the phenomenon of bubbling.

We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit. An important feature of our approach is that the required assumptions for which our results hold are met by a general class of maps, improving in this way some previous results. We discuss the applicability of our scheme to some models of population dynamics, and give numerical simulations to illustrate our analytical results.

We address the stability properties in a non-autonomous difference equation where f is continuous, and the zero solution is assumed to be the unique equilibrium. We focus our discussion on two techniques motivated by stability results for functional differential equations (FDEs) that proved recently to be useful in the frame of difference equations too. The first one involves the use of discrete inequalities and monotonicity arguments, and it is inspired by the so-called Halanay inequality; the second one is based on the well-known 3/2 stability results for FDEs. We give further insight into the simple ideas that are behind these methods, prove some new results and show applications and open problems.

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 study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.

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.

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

EXISTENCE OF PERIODIC SOLUTIONS FOR FUNCTIONAL EQUATIONS WITH PERIODIC DELAY

We address the problem of global periodicity in discrete dynamical systems generated by rational maps in or . Our main results show that for a wide family of such maps, this problem may be reduced to the analysis of a related matrix equation. We use this fact to estimate the number of possible minimal periods in globally periodic maps of this class when all the involved coefficients are rational.