Nonlinear population dynamics: models, experiments and data.
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Animals↗
Biomedical subjects
J M Cushing
Publications and source records attributed to J M Cushing.
Resonant population cycles in temporally fluctuating habitats.
Experiments with the flour beetle Tribolium have revealed that animal numbers were larger in cultures grown in a periodically fluctuating volume of medium than in cultures grown in a constant volume of the same average size. In this paper we derive and analyze a discrete stage-structured mathematical model that explains this phenomenon as a kind of resonance effect. Habitat volume is incorporated into the model by the assumption that all rates of cannibalism (larvae on eggs, adults on eggs and pupae) are inversely proportional to the volume of the culture medium. We tested this modeling assumption by conducting and statistically analyzing laboratory experiments. For parameter estimates derived from experimental data, our model indeed predicts, under certain circumstances, a larger (cycle-average) total population abundance when the habitat volume periodically fluctuates than when the habitat volume is held constant at the average volume. The model also correctly predicts certain phase relationships and transient dynamics observed in data. The analyses involve a thorough integration of mathematics, statistical methods, biological details and experimental data.
Animals↗
A simple model of cannibalism.
A simple nonlinear discrete model is derived for the dynamics of a two-age class population consisting of juveniles and adults that includes cannibalism of juveniles by adults. The model is investigated analytically and numerically. It is shown how even this very simple model, by incorporating the negative and positive feedbacks due to cannibalism, can account for several important phenomena concerning the dynamics of cannibalistic populations that have been discussed and studied in the literature. These include the possibilities that the practice of cannibalism can (1) in certain circumstances be a form of self-regulation that promotes stable equilibration, while in other circumstances it can lead to population oscillations; (2) result in a viable population in circumstances when its absence would otherwise result in extinction; and (3) be the source of multiple stable equilibria and hysteresis effects.
Age Factors↗
Equilibria in systems of interacting structured populations.
The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.
Animals↗
Periodic Lotka-Volterra competition equations.
The Lotka-Volterra competition equations with periodic coefficients derived from the MacArthur-Levins theory of a one-dimensional resource niche are studied when the parameters are allowed to oscillate periodically in time. Specifically, niche positions and widths, resource availability and resource consumption rates are allowed small amplitude periodicities around a specified mean value. Two opposite cases are studied both analytically and numerically. First only resource consumption rates are allowed to oscillate while niche dimensions and resource availability are held constant. The resulting oscillations in population densities and the strength of the system stability as they depend upon crucial relative phase and amplitude differences between the species' consumption rates are studied. This leads to a clear notion of "temporal niche" and of the effects that such oscillations can have on competitive coexistence. Secondly, all system parameters are allowed to oscillate, although the oscillatory consumption rates are assumed identical for both species. The effects on the population density oscillations and their averages are studied and the "best" choice of the common, periodic resource consumption rate for these two "identical" species competing for similar (even identical) niches is considered.
Animals↗
Equilibria in structured populations.
The existence of a stable positive equilibrium state for the density rho of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of of nontrivial equilibrium pairs (n, rho) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.
Animals↗
A predator prey model with age structure.
A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects it fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and de-stabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation period m of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for small m or are destabilized as m decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.
Aging↗
Model stability and instability in age structured populations.
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Age Factors↗
Time delays in single species growth models.
A general model is considered for the growth of a single species population which describes the per unit growth rate as a general functional of past population sizes. Solutions near equilibrium are studied as function of epsilon = 1/b, the reciprocal of the inherent per unit growth rate b of the population in the absense of any density constraints. Roughly speaking, it is shown that for large epsilon the equilibrium is asymptotically stable and that for epsilon small the solutions show divergent oscillations around the equilibrium. In the latter case a first order approximation is obtained by means of singular perturbation methods. The results are illustrated by means of a numerically integrated delay-logistic model.
Models, Theoretical↗
Predator prey interactions with time delays.
A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.
Animals↗