Continuous-time Population Models Research Articles

Positivity and dynamics preserving discretization schemes for nonlinear evolution equations

Discretization of a continuous-time system of dierential equations becomes inevitable due to the lack of analytical solutions. Standard discretization techniques, however, have many things that could be improved, e.g., the positivity of the solution and dynamic consistency may be lost, and stability and convergence may depend on the step length. A nonstandard nite dierence (NSFD) scheme is sometimes used to avoid inconsistencies. There are two fundamental issues regarding the construction of NSFD models. First, how to construct the denominator function of the discrete rst-order derivative? Second, how to discretize the nonlinear terms of a given dierential equation with nonlocal terms? We dene here a uniform technique for nonlocal discretization and construction of denominator function for NSFD models. We have discretized a couple of highly nonlinear continuous-time population models using these consistent rules. We give analytical proof in each case to show that the proposed NSFD model has identical dynamic properties to the continuous-time model. It is also shown that each NSFD system is positively invariant, and its dynamics do not depend on the step size. Numerical experiments have also been performed in favour of such claims.

Stability and bifurcation analyses of a discrete Lotka–Volterra type predator–prey system with refuge effect

In this paper, we discuss the complex dynamical behavior of a discrete Lotka–Volterra type predator–prey model including refuge effect. The model considered is obtained from a continuous-time population model by utilizing the forward Euler method. First of all, we nondimensionalize the system to continue the analysis with fewer parameters. And then, we determine the fixed points of the dimensionless system. We investigate the dynamical behavior of the system by performing the local stability analysis for each fixed point, separately. Moreover, we analytically show the existence of flip and Neimark–Sacker bifurcations at the positive fixed point by applying the normal form theory and the center manifold theorem. Bifurcation analyses are carried out by choosing the integral step size as a bifurcation parameter. In addition, we perform numerical simulations to support and extend the analytical results. All these analyses have been done for the models with and without the refuge effect to examine the effect of refuge on the dynamics. We have concluded that the refuge has significant role on the dynamical behavior of a discrete system. Furthermore, numerical simulations underline that the large integral step size causes the chaotic behavior.

Growth rate and basic reproduction number for population models with a simple periodic factor

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.

Chaos in a Three‐Species Food Chain with a Lower Bound on the Bottom Population

Hastings and Powell (1991) present a simple, differential-equation model of a food chain where a predator population (P), feeds on a consumer population (C), which in turn feeds on a resource population (R). They observe that this model can exhibit chaotic dynamics. McCann and Yodzis (1994a), reformulate the model and use the method of Yodzis and Innes (1992) in order to produce plausible combinations of parameter values. They conclude that the parameters chosen by Hastings and Powell (1991) are biologically extreme. Moreover, they suggest that the chaos shown by the model with these parameter values is unlikely to be persistent in a real system because it involves the predator population (P) dropping to very low levels. However, McCann and Yodzis (1994a) produce a range of more plausible parameter values and demonstrate the existence of chaos for a wide range of these values. This chaos is shown to persist, with P being bounded comfortably away from zero. Simple, difference-equation models of single populations have frequently been shown to be capable of exhibiting chaotic behavior (e.g., May 1976). However, McCallum (1992) demonstrates that the addition of a small immigration term, or a refuge that sets a nonzero minimum level for the population, greatly reduces the propensity of such models to behave chaotically. This work has been expanded by others (e.g., GonzalezAndujar and Perry 1993a, b, Hastings 1993, Stone 1993, Ruxton 1994). McCallum (1992) speculates that similar avoidance of chaos could occur in differentialequation models. However this has not yet been shown. Here, I use a modification of the model of McCann and Yodzis (1994a) to investigate if the imposition of a population floor affects continuous-time population models in a way similar to that of discrete-time models, namely by reducing the propensity to exhibit chaos. If such an effect is found, then this would suggest that

Analysis of a distributed-parameter population model based on physiological age

Classification of individuals in a population by a measure of physiological, rather than chronological, age is often desirable, but continuous-time population models based on physiological age have yielded little to analysis. For the partial differential equation model developed here, analytical solutions can be obtained when fertility, mortality, and individual growth rates are assumed time-invariant. The solutions permit a generalization of Lotka's classical stability results to physiological-age distributions. The shape of the stable physiological-age distribution is derived and compared with stable chronological-age profiles. The resulting equations provide a model for the “stacking up”, in physiological age, of individuals in the population. This phenomenon is often reflected in the size-frequency distribution of fish and invertebrate populations.

A continuous-time population model with poisson recruitment

A continuous-time model of a multigrade system is developed, which includes Poisson arrivals, interaction between grades and a leaving process. It therefore constitutes a continuous-time analogue of Pollard's hierarchical population model with Poisson recruitment. An expression is found for the first and second moments of grade size at any time. A general formulation of the joint probability generating function of the numbers in each grade is given, and the limiting distribution of grade size is shown to be Poisson.

A continuous-time population model with poisson recruitment

A continuous-time model of a multigrade system is developed, which includes Poisson arrivals, interaction between grades and a leaving process. It therefore constitutes a continuous-time analogue of Pollard's hierarchical population model with Poisson recruitment. An expression is found for the first and second moments of grade size at any time. A general formulation of the joint probability generating function of the numbers in each grade is given, and the limiting distribution of grade size is shown to be Poisson.