Dynamical consequences of harvest in discrete age-structured population models

The role of harvest in discrete age-structured one-population models has been explored. Considering a few age classes only, together with the overcompensatory Ricker recruitment function, we show that harvest acts as a weak destabilizing effect in case of small values of the year-to-year survival probability P and as a strong stabilizing effect whenever the survival probability approaches unity. In the latter case, assuming n=2 age classes, we find that harvest may transfer a population from the chaotic regime to a state where the equilibrium point (x₁*, x₂*) becomes stable. However, as the number of age classes increases (which acts as a stabilizing effect in non-exploited models), we find that harvest acts more and more destabilizing, in fact, when the number of age classes has been increased to n=10, our finding is that in case of large values of the survival probabilities, harvest may transfer a population from a state where the equilibrium is stable to the chaotic regime, thus exactly the opposite of what was found in case of n=2. On the other hand, if we replace the Ricker relation with the generalized Beverton and Holt recruitment function with abruptness parameter larger than 2, several of the conclusions derived above are changed. For example, when n is large and the survival probabilities exceed a certain threshold, the equilibrium will always be stable.Revised version: 18 September 2003     

A unifying framework for chaos and stochastic stability in discrete population models

 In this paper we propose a general framework for discrete time one-dimensional Markov population models which is based on two fundamental premises in population dynamics. We show that this framework incorporates both earlier population models, like the Ricker and Hassell models, and experimental observations concerning the structure of density dependence. The two fundamental premises of population dynamics are sufficient to guarantee that the model will exhibit chaotic behaviour for high values of the natural growth and the density-dependent feedback, and this observation is independent of the particular structure of the model. We also study these models when the environment of the population varies stochastically and address the question under what conditions we can find an invariant probability distribution for the population under consideration. The sufficient conditions for this stochastic stability that we derive are of some interest, since studying certain statistical characteristics of these stochastic population processes may only be possible if the process converges to such an invariant distribution. Received 15 May 1995; received in revised form 17 April 1996     

Stability of discrete age-structured and aggregated delay-difference population models

Sufficiency conditions for local stability are derived for a class of density dependent Leslie matrix models. Four of the recruitment functions in common use in fisheries management are then considered. In two of these oscillating instability can never occur (Beverton and Holt and Cushing forms). In the other two (Deriso-Schnute and Shepherd forms) undamped oscillations are possible within the region of parameter space described here. An algorithm is developed for calculating necessary and sufficient local stability conditions for a simplified form of the general age-structured model. The complete spectrum of stability states (monotonic stability; monotonic instability; oscillating-stable; oscillating-unstable) and the bifurcation periods are given for selected examples of this model. The examples cover a large portion of the parameter space of interest in resource management. It is shown that in perfectly deterministic systems which are observed with error, oscillating instabilities may be missed, and such systems could be erroneously assumed to be stable.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Stochastic density dependence in population size of a benthic clonal invertebrate: the regulating role of fission

The influence of environmental variation on the demography of clonal organisms has been poorly studied. I utilise a matrix model of the population dynamics of the intertidal zoanthid Palythoa caesia to examine how density dependence and temporal variation in demographic rates interact in regulating population size. The model produces realistic simulations of population size, with erratic fluctuations between soft lower and upper boundaries of approximately 55 and 90% cover. Cover never exceeds the maximum possible of 100%, and the population never goes to extinction. A sensitivity analysis indicates that the model’s behaviour is driven by density dependence in the fission of large colonies to produce intermediate sized colonies. Importantly, there is no density-dependent mortality in the model, and density dependence in recruitment, while present, is unimportant. Thus it appears that the main demographic processes which are considered to regulate population size in aclonal organisms may not be important for clonal species. Received: 18 August 1999 / Accepted: 29 October 1999     

Effects of delay,truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models

Understanding effects of hypotheses about reproductive influences, reproductive schedules and the model mechanisms that lead to a loss of stability in a structured model population might provide information about the dynamics of natural population. To demonstrate characteristics of a discrete time, nonlinear, age structured population model, the transition from stability to instability is investigated. Questions about the stability, oscillations and delay processes within the model framework are posed. The relevant processes include delay of reproduction and truncation of lifetime, reproductive classes, and density dependent effects. We find that the effects of delaying reproduction is not stabilizing, but that the reproductive delay is a mechanism that acts to simplify the system dynamics. Density dependence in the reproduction schedule tends to lead to oscillations of large period and towards more unstable dynamics. The methods allow us to establish a conjecture of Levin and Goodyear about the form of the stability in discrete Leslie matrix models.This research was supported in part by the US Environmental Protection Agency under cooperation agreement CR-816081     

Interspecific competition among macroparasites in a density-dependent host population

 We analyze the dynamics of a community of macroparasite species that share the same host. Our work extends an earlier framework for a host species that would grow exponentially in the absence of parasitism, to one where an uninfected host population is regulated by factors other than parasites. The model consists of one differential equation for each parasite species and a single density-dependent nonlinear equation for the host. We assume that each parasite species has a negative binomial distribution within the host and there is zero covariance between the species (exploitation competition). New threshold conditions on model parameters for the coexistence and competitive exclusion of parasite species are derived via invadibility and stability analysis of corresponding equilibria. The main finding is that the community of parasite species coexisting at the stable equilibrium is obtained by ranking the species according t! o th e minimum host density H ^* above which a parasite species can grow when rare: the lower H ^* , the higher the competitive ability. We also show that ranking according to the basic reproduction number Q ₀ does not in general coincide with ranking according to H ^* . The second result is that the type of interaction between host and parasites is crucial in determining the competitive success of a parasite species, because frequency-dependent transmission of free-living stages enhances the invading ability of a parasite species while density-dependent transmission makes a parasite very sensitive to other competing species. Finally, we show that density dependence in the host population entails a simplification of the portrait of possible outcomes with respect to previous studies, because all the cases resulting in the exponential growth of host and parasite populations are eliminated.. Received: 24 June 1996 / Revised version: 28 April 1998     

Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semi-stable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the origin’s basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

Global stability in consumer-resource cascades

 Models of population growth in consumer-resource cascades (serially arranged containers with a dynamic consumer population, v, receiving a flow of resource, u, from the previous container) with a functional response of the form h(u/v ^b ) are investigated. For b∈[0, 1], it is shown that these models have a globally stable equilibrium. As a result, two conclusions can be drawn: (1) Consumer density dependence in the functional or in the per-capita numerical response can result in persistence of the consumer population in all containers. (2) In the absence of consumer density dependence, the consumer goes extinct in all containers except possibly the first. Several variations of this model are discussed including replacing discrete containers by a spatial continuum and introducing a dynamic resource. Received 25 February 1995 / received in revised form 27 July 1995     

On the formulation and analysis of general deterministic structured population models I. Linear Theory

—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R ₀ and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step. Received 26 July 1996; received in revised form 3 September 1997     

Excitability of an age-structured plankton ecosystem

 We adapt a simple two-component model of a plankton ecosystem to account for the life spans of individual predatory organisms. We investigate the system’s short-term dynamics, in particular its excitability, and its long-term dynamics, and show how both can be highly sensitive to initial conditions. We discover that this effect is enhanced by imposing age structure on the system. Received: 21 June 1996 / Revised version: 19 October 1998     

Genealogy and subpopulation differentiation under various models of population structure

 The structured coalescent is used to calculate some quantities relating to the genealogy of a pair of homologous genes and to the degree of subpopulation differentiation, under a range of models of subdivided populations and assuming the infinite alleles model of neutral mutation. The classical island and stepping-stone models of population structure are considered, as well as two less symmetric models. For each model, we calculate the Laplace transform of the distribution of the coalescence time of a pair of genes from specified locations and the corresponding mean and variance. These results are then used to calculate the values of Wright’s coefficient F _ST , its limit as the mutation rate tends to zero and the limit of its derivative with respect to the mutation rate as the mutation rate tends to zero. From this derivative it is seen that F _ST can depend strongly on the mutation rate, for example in the case of an essentially one-dimensional habitat with many subpopulations where gene flow is restricted to neighbouring subpopulations. Received: 1 October 1997 / Revised version: 15 March 1998     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Strong density-dependent survival and recruitment regulate the abundance of a coral reef fish

Debate on the control of population dynamics in reef fishes has centred on whether patterns in abundance are determined by the supply of planktonic recruits, or by post-recruitment processes. Recruitment limitation implies little or no regulation of the reef-associated population, and is supported by several experimental studies that failed to detect density dependence. Previous manipulations of population density have, however, focused on juveniles, and there have been no tests for density-dependent interactions among adult reef fishes. I tested for population regulation in Coryphopterus glaucofraenum, a small, short-lived goby that is common in the Caribbean. Adult density was manipulated on artificial reefs and adults were also monitored on reefs where they varied in density naturally. Survival of adult gobies showed a strong inverse relationship with their initial density across a realistic range of densities. Individually marked gobies, however, grew at similar rates across all densities, suggesting that density-dependent survival was not associated with depressed growth, and so may result from predation or parasitism rather than from food shortage. Like adult survival, the accumulation of new recruits on reefs was also much lower at high adult densities than at low densities. Suppression of recruitment by adults may occur because adults cause either reduced larval settlement or reduced early post-settlement survival. In summary, this study has documented a previously unrecorded regulatory mechanism for reef fish populations (density-dependent adult mortality) and provided a particularly strong example of a well-established mechanism (density-dependent recruitment). In combination, these two compensatory mechanisms have the potential to strongly regulate the abundance of this species, and rule out the control of abundance by the supply of recruits.     

On density and extinction in continuous population models

Survival analyses, investigations of extinction and persistence, are executed for populations represented by a nonautonomous differential equation model. The population is assumed governed by density dependent and time varying density independent demographic parameters. While traditional approaches to extinction postulate extinction on an infinite time horizon and at zero abundance level, survival analysis is developed not only for this traditional setting but also on a finite time horizon and at a nonzero threshold level. A main conclusion is that extinction of a temporally stressed population is determined by a totality of density independent and density dependent factors.     

Global asymptotic stability of density dependent integral population projection models

Many stage-structured density dependent populations with a continuum of stages can be naturally modeled using nonlinear integral projection models. In this paper, we study a trichotomy of global stability result for a class of density dependent systems which include a Platte thistle model. Specifically, we identify those systems parameters for which zero is globally asymptotically stable, parameters for which there is a positive asymptotically stable equilibrium, and parameters for which there is no asymptotically stable equilibrium.     

Segregation and mixing in degenerate diffusion in population dynamics

 We study the qualitative properties of degenerate diffusion equations used to describe dispersal processes in population dynamics. For systems of interacting populations, the forms of the diffusion models used determine if the population will intermix or remain disjoint (segregated). The dynamics and stability of segregation boundaries between competing populations is analyzed. General population models with segregation and mixing interactions are derived and connections to behavior in fluid mechanical systems are addressed. Received 19 January 1996; received in revised form 4 April 1996     

Plant litter feedback and population dynamics in an annual plant,<Emphasis Type="Italic"> Cardamine pensylvanica</Emphasis>

The presence of litter has the potential to alter the population dynamics of plants. In this paper, we explore the effects of litter on population dynamics using a simple experimental laboratory system with populations of the annual crucifer, Cardamine pensylvanica. Using a factorial experiment with four densities and three litter levels, we determined the effect of litter on biomass and plant fecundity, and the life stages responsible for these changes in yield. Although litter had significant effects on seed germination and on seedling survivorship, we show, using a population dynamics model, that these effects were not demographically significant. Rather, the potential effect of litter on population dynamics resulted almost entirely from its effect on biomass. Persistent litter suppressed plant biomass and apparently removed the direct density effect present in the absence of litter. Thus, litter changed the shape of the recruitment curve from slightly humped to asymptotic. In addition to changing the shape of the recruitment curve, litter reduced the carrying capacity of the populations. Thus, the population dynamics model indicated that not all statistically significant responses were dynamically significant. Given the potential complexity of litter effects, simple population models provide a powerful tool for understanding the potential consequences of short-term responses. Received: 8 September 1999 / Accepted: 5 April 2000