Estimating population growth rates from stochastic Leslie matrices

Stochastic versions of exponential growth models predict that even when r or λ values calculated from mean vital statistics indicate exponential growth, most of the individual populations may become extinct. Several recent papers have considered this problem and some misunderstanding has arisen due to the difference between mathematical expectation of population size and most likely course of population growth. We replicated Boyce's (1977, 1979) simulations of population growth with age structure and a single randomly varying vital statistic, and reconciled some of these differences. Mean number can be projected using the dominant eigenvalue of the mean Leslie matrix, but the modal number may be considerably lower. We compared several measures of the rate of growth of the geometric mean or median of numbers and conclude that Tuljapurkar's α is an acceptable measure.     

Asymptotic properties of infinite Leslie matrices

The stable population theory is classically applicable to populations in which there is a maximum age after which individuals die. Demetrius [1972. On an infinite population matrix. Math. Biosci. 13, 133-137] extended this theory to infinite Leslie matrices, in which the longevity of individuals is potentially infinite. However, Demetrius had to assume that the survival probability per time step tends to 0 with age. We generalise here the conditions of application of the stable population theory to infinite Leslie matrix models and apply these results to two examples, including or not senescence.     

Bias in estimating the Malthusian parameter for Leslie matrices

For Leslie matrices of order 3 × 3 or larger, conditions for concavity or convexity of the Malthusian parameter in each of the entries in the matrix are given. Both cases are possible so it follows that the expected population growth rate computed from a Leslie matrix whose entries are random variables can be either smaller or larger than the growth rate computed from the expected value of the matrix. Boyce [(1977) Theor. Pop. Biol.12] showed that in the 2 × 2 case this bias is always positive; we give a numerical example illustrating the magnitude of the bias in this case, and compare it with the sampling error of the parameter for the same example.     

Primitivity of the product of two Leslie matrices

Necessary and sufficient conditions for primitivity of a product of two Leslie matrices are given. Such a product could be used in modeling the growth of a population governed alternately by two different sets of fertility and survival parameters.     

Properties of the Leslie population matrix

A simple similarity transformation of the Leslie matrix renders certain properties obvious. In particular the characteristic polynomial, characteristic vectors and principal vectors can be explicitly written out. Bounds for the dominant root are given.     

Random transitions,size control,and inheritance in cell population dynamics

A general model of cell population dynamics is derived and analyzed. The model uses the continuous structure variables age and size, and thus distinguishes individual cells with respect to such properties as cycle length and division size. The model allows the occurrence of random transitions as cells progress through the cell cycle, the control of cell size upon cell cycle events, and the inheritance of properties from mother to daughter cells. The concepts of asynchronous exponential growth, α-curves, β-curves, mother-daughter transit time correlations, and sister-sister transit time correlations are formalized. The existence and uniqueness of solutions to the model is proved.     

Random matrices and neural networks

In connection with some problems that arise in the study of neural networks random matrices are considered and the probability for them to have certain rank is investigated. Two models are studied in a simple-minded approach to problems of this type.On leave of absence from the Institute for Mathematical Sciences, Madras (India).     

Projection matrices in population biology

Projection matrix models are widely used in population biology to project the present state of a population into the future, either as an attempt to forecast population dynamics, or as a way to evaluate life history hypotheses. These models are flexible and mathematically relatively easy. They have been applied to a broad range of plants and animals. The asymptotic properties of projection matrices have clearly defined biological interpretations, and the analysis of the effects of perturbations on these asymptotic properties offers new possibilities for comparative life history analysis. The connection between projection matrix models and the secondary theorem of natural selection opens life cycle phenomena to evolutionary interpretation.     

Raising Leslie matrices to powers: a method and applications to demography

An illustrative method, labelled Strip and Mask, to raise a Leslie matrix to powers is introduced. Starting from a recent article in this journal, the Strip and Mask method is utilized to determine the primitivity pattern of a Leslie matrix, and to discuss some properties of the corresponding population model.     

Harvesting in a resource dependent age structured Leslie type population model

We analyse the effect of harvesting in a resource dependent age structured population model, deriving the conditions for the existence of a stable steady state as a function of fertility coefficients, harvesting mortality and carrying capacity of the resources. Under the effect of proportional harvest, we give a sufficient condition for a population to extinguish, and we show that the magnitude of proportional harvest depends on the resources available to the population. We show that the harvesting yield can be periodic, quasi-periodic or chaotic, depending on the dynamics of the harvested population. For populations with large fertility numbers, small harvesting mortality leads to abrupt extinction, but larger harvesting mortality leads to controlled population numbers by avoiding over consumption of resources. Harvesting can be a strategy in order to stabilise periodic or quasi-periodic oscillations in the number of individuals of a population.     

The problem of time unit in Leslie’s population model

The conditions that will allow the lumping together of several age classes in the Leslie model are investigated. We show that if the lumping is to be valid for all population distributions, then the parameters of the model must be periodic. Lumping is valid when the population is in equilibrium, but equilibrium should be tested before the model is lumped.     

Optimal population stabilization and control using the Leslie matrix model

We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class ofshifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained opitmization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.     

Small population effects in stochastic population dynamics

The discreteness of units of small populations can produce large fluctuations from a classical continuous representation, especially when null populations play a crucial role. These belong to what are here referred to as emergent and evanescent species. A few model biological systems are introduced in which this is the case, as well as a toy model that suggests a path to avoid the associated mathematical complexities. The corresponding division into null and non-null population sectors is carried out to leading order for the model systems, with promising results. Supported in part by DOE Office of Basic Science, Chemical Division. Reported at SMB03, the August 2003 meeting of the Society for Mathematical Biology.     

Dose-structured population dynamics

Applied population dynamics modeling is relied upon with increasing frequency to quantify how human activities affect human and non-human populations. Current techniques include variously the population's spatial transport, age, size, and physiology, but typically not the life-histories of exposure to other important things occurring in the ambient environment, such as chemicals, heat, or radiation. Consequently, the effects of such 'abiotic' aspects of an ecosystem on populations are only currently addressed through individual-based modeling approaches that despite broad utility are limited in their applicability to realistic ecosystems [V. Grimm, Ten years of individual-based modeling in ecology: what have we learned and what could we learn in the future? Ecol. Model. 115 (1999) 129-148][1]. We describe a new category of population dynamics modeling, wherein population dynamical states of the biotic phases are structured on dose, and apply this framework to demonstrate how chemical species or other ambient aspects can be included in population dynamics in three separate examples involving growth suppression in fish, inactivation of microorganisms with ultraviolet irradiation, and metabolic lag in population growth. Dose-structuring is based on a kinematic approach that is a simple generalization of age-structuring, views the ecosystem as a multi-component mixture with reacting biotic/abiotic components. The resulting model framework accommodates (a) different memories of exposure as in recovery from toxic ambient conditions, (b) differentiation between exogenous and endogenous sources of variation in population response, and (c) quantification of acute or sub-acute effects on populations arising from life-history exposures to abiotic species. Classical models do not easily address the very important fact that organisms differ and have different experiences over their life cycle. The dose structuring is one approach to incorporate some of these elements into the existing structures of the classical models, while retaining many of the features (and other limitations) of classical models.     

Linearized oscillations in population dynamics

A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.