Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.     

Allee effects in a discrete-time SIS epidemic model with infected newborns

We extend the framework of Rios-Soto et al. (Contemporary Mathematics, 2006, 410, 297) to include both compensatory (contest competition) and overcompensatory (scramble competition) population dynamics with and without the Allee effect. We compute the basic reproductive number ?₀, and use it to predict the (uniform) persistence or extinction of the infective population, where the population dynamics are compensatory and the Allee effect is either present or absent. We also explore the relationship between the demographic equation and the epidemic process, where the total population dynamics are overcompensatory. In particular, we show that the demographic dynamics drive both the susceptible and infective dynamics. This is in contrast to the recent observations of Franke and Yakubu, that the demographic dynamics can be chaotic while the infective dynamics are oscillatory and non-chaotic in periodically-forced SIS epidemic models (Mathematical Biosciences, 2006, 204, 68).     

Weak Allee effects and species coexistence

In this article, we study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

On chaos, transient chaos and ghosts in single population models with Allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)^∞ occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects.     

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

Dynamic complexities in predator–prey ecosystem models with age-structure for predator

Natural populations, whose generations are non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, in former studies most of the investigations of complex population dynamics were mainly concentrated on single populations instead of higher dimensional ecological systems. This paper reports a recent study on the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction based on age-structure of predator. The complexities include (a) non-unique dynamics, meaning that several attractors coexist; (b) antimonotonicity; (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties, consisting of pattern of self-similarity and fractal basin boundaries; (d) intermittency; (e) supertransients; and (f) chaotic attractors.     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

Bio-economics of a renewable resource subjected to strong Allee effect

In this article bio-economics of a renewable resource that is subjected to strong Allee effect (multiplicative Allee effect) is investigated from sole owner perspective. The considered optimal harvesting problem has been solved using Pontryagin maximum principle. The control problem admits multiple singular equilibrium solutions in contrast to the case where the growth of the resource is of compensatory nature. Thus the choice of optimal singular solution and the nature of associated approach paths make the problem pertinent and interesting.     

Dynamical complexity induced by Allee effect in a predator–prey model

In this paper, we investigate the complex dynamics induced by Allee effect in a predator–prey model. For the non-spatial model, Allee effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee effect, the model is bistable. For the spatial model, without Allee effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee effect is not simple, but rich and complex.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Periodic solutions of some models with strong Allee effects

In this paper, the authors obtain sufficient conditions for the existence of two positive periodic solutions to models representing the dynamics of a renewable resource that is subject to a strong Allee effect. The Leggett-Williams multiple fixed point theorem is used to prove the results.     

From Individuals to Populations: A Symbolic Process Algebra Approach to Epidemiology

Is it possible to symbolically express and analyse an individual-based model of disease spread, including realistic population dynamics? This problem is addressed through the use of process algebra and a novel method for transforming process algebra into Mean Field Equations. A number of stochastic models of population growth are presented, exploring different representations based on alternative views of individual behaviour. The overall population dynamics in terms of mean field equations are derived using a formal and rigorous rewriting based method. These equations are easily compared with the traditionally used deterministic Ordinary Differential Equation models and allow evaluation of those ODE models, challenging their assumptions about system dynamics. The utility of our approach for epidemiology is confirmed by constructing a model combining population growth with disease spread and fitting it to data on HIV in the UK population. This work was supported by EPSRC through a Doctoral Training Grant (CM, from 2004–2007), and through System Dynamics from Individual Interactions: A process algebra approach to epidemiology (EP/E006280/1, all authors, 2007–2010).     

Exact travelling wave solutions of reaction-diffusion models of fractional order

Reaction-diffusion models are used in different areas of chemistry problems. Also, coupled reaction-diffusion systems describing the spatio- temporal dynamics of competition models have been widely applied in many real world problems. In this paper, we consider a coupled fractional system with diffusion and competition terms in ecology, and reaction-diffusion growth model of fractional order with Allee effect describing and analyzing the spread dynamic of a single population under different dispersal and growth rates. Finding the exact solutions of such models are very helpful in the theories and numerical studies. Exact traveling wave solutions of the above reaction-diffusion models are found by means of the $Q$-function method. Moreover, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviours.     

Julia sets for the gamma recursion in nonlinear psychophysics

Julia sets for the map za(z–ie)(1–z)(z+ie) are illustrated for some attractors of interest. This work extends previous analyses of the cubic complex polynomial and considers dynamics in regions which may be associated with the modelling of the results of overload in sensory inputs.Work supported by grant No. A7881403 (1988) from the Australian Research Council.     

Dynamics of a single species evolutionary model with Allee effects

We investigate the evolutionary outcomes of a single species population subject to Allee effects within the framework of a continuous strategy evolutionary game theory (EGT) model. Our model assumes a single trait creates a phenotypic trade-off between carrying capacity (i.e., competition) and predator evasion ability following a Gaussian distribution. This assumption contributes to one of our interesting findings that evolution prevents extinction even when population exhibits strong Allee effects. However, the extinction equilibrium can be an ESS under some special distributions of anti-predation phenotypes. The ratio of variation in competition and anti-predation phenotypes plays an important role in determining global dynamics of our EGT model: (a) evolution may suppress strong Allee effects for large values of this ratio; (b) evolution may preserve strong Allee effects for small values of this ratio by generating a low density evolutionary stable strategy (ESS) equilibrium which can serve as a potential Allee threshold; and (c) intermediate values of this ratio can result in multiple ESS equilibria.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.