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.     

Mathematical analysis of the asymptotic behavior of the Leslie population matrix model

LetL be a Leslie population matrix. Leslie (1945) and others have shown that the matrixL has a leading positive eigenvalueλ ₀ and that in general: (1) $$\mathop {\lim }\limits_{t \to \infty } \frac{{L^t X}}{{\lambda _0^t }} = \gamma X_{\lambda _0 } $$ whereX _{λ 0} is an eigenvector corresponding toλ ₀,X is any initial population vector, and γ is a scalar quantity detormined byX. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms ofλ ₀ and the entries of the matrixL the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is nonzero. To stabilize a population, the matrixL must be adjusted so thatλ ₀=1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stanbilization which yields as a byproduct an easily accessible scalar measure ofL's tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.     

A density-dependent Leslie matrix model

A density-dependent Leslie matrix model introduced in 1948 by Leslie is mathematically analyzed. It is shown that the behavior is similar to that of the constant Leslie matrix. In the primitive case, the density-dependent Leslie matrix model has an asymptotic distribution corresponding to the logistic equation. However, in the imprimitive case, the asymptotic distribution is periodic, with period depending on the imprimitivity index.     

Random Leslie matrices in population dynamics

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3 × 3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.     

On the history of populations governed by a Leslie matrix

Even though the Leslie matrix is usually singular, there is a subspace on which is has an inverse. In addition, there is a projection into that subspace which preserves certain age classes. These two facts are combined to provide a model for the history of a population whose future is predicted by a Leslie matrix. It has the advantage of being composed of easily calculated matrices. The relation of this model to a backward projection method of Greville and Keyfitz is discussed and some other backward projection functions are proposed.     

Computerization of a population projection model based on generalized age-dependent branching processes: a connection with the Leslie matrix

The population projection model based on generalized age-dependent branching processes developed by Mode and Busby (1981) involves the solution of a large number of renewal type equations. It is shown that these equations may be solved recursively. Such a solution has two implications. One is that the projection model may be very efficiently computerized. Second, the recursive algorithm developed has striking similarities to two traditional methods of population projection used by demographers: the Leslie matrix and cohort component methods. The results presented here associate traditional projection techniques with the theory of age-dependent branching processes.     

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.     

Single-class orbits in nonlinear Leslie matrix models for semelparous populations

The dynamics of a general nonlinear Leslie matrix model for a semelparous population is investigated. We are especially concerned with the attractivity of the single-class state, in which all but one cohort (or year-class) are missing. Our result shows that the single-class state is attractive if inter-class competition is severe. Conversely, if intra-class competition is severe, the single-class state is repelling. Numerical investigations show that all classes do not necessarily coexist even if the single-class state is repelling.     

An extension of the Leslie matrix model to include a variable immature period

An extension to the Leslie matrix is presented in which the age of transformation from immature to adult has a log—normal distribution. The major effect of this is shown to be on the second largest eigenvalue. The ratio of the largest to the second largest eigenvalue |λ₁/λ₂|, which is an index of the rate of approach to the stable age distribution, is greater in the new model, even though the value of λ₁ is effectively the same. The differences in the models are most pronounced where the population is subjected to a harvesting regime.     

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. The level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.     

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.     

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.     

Modelling population growth in stage-grouped organisms: a simple extension to the Leslie model

A simple method for decomposing a population's stage grouping into the underlying age structure is described. The population's dynamics can then be predicted using standard age-structured models, such as Leslie's matrix model. The method overcomes objections to previous attempts to use Leslie's procedure for modelling population growth in stage-grouped organisms. A hypothetical example is used to illustrate the new technique.     

On the construction of the population projection matrix for a population grouped in unequal stages.

It is shown how it is possible to construct a projection matrix for a population grouped in unequal stages directly from a known projection matrix for the population grouped in equal age intervals.     

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     

Evaluation of the population dynamics of the forage legume (Lotus corniculatus), using matrix population models

The population dynamics of perennial crop plants are influenced by numerous factors, including management practices. Conditions in the field vary from year to year, and matrix population models are useful for evaluating population behaviour in relation to environmental variability. In Missouri, the stand persistence of birdsfoot trefoil ( Lotus corniculatus ), a perennial legume, is often limited by disease and poor seed production. A stage-based, matrix population model was developed to evaluate the population dynamics of birdsfoot trefoil in relation to clipping treatment. The plant growth stages represented in the model were seeds, seedlings, mature vegetative and reproductive plants. Two phases of population growth were evaluated in clipped and unclipped stands. Establishment-phase populations were characterized by relatively high mortality and low reproduction. Elasticity analysis indicated that growth of these populations was most sensitive to the survival of vegetative plants. Mature vegetative plants and seeds comprised the majority of surviving individuals in clipped and unclipped populations, respectively; however, establishment-phase populations under both management treatments tended toward extinction. Populations in the post-establishment phase of growth were characterized by relatively low mortality and high reproduction. Population growth in this phase of growth was most sensitive to seed production, and most individuals in these populations were at the seed stage.