Continuous Population Models for Single Species
The increasing study of realistic mathematical models in ecology (basically the study of the relation between species and their environment) is a reflection of their use in helping to understand the dynamic processes involved in such areas as predator-pre
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The increasing study of realistic mathematical models in ecology (basically the study of the relation between species and their environment) is a reflection of their use in helping to understand the dynamic processes involved in such areas as predator-prey and competition interactions, renewable resource management, evolution of pesticide resistant strains, ecological control of pests, multi-species societies, plant-herbivore systems and so on. The continually expanding list of applications is extensive. There are also interesting and useful applications of single species models in the biomedical sciences: in Section 1.5 we discuss two practical examples of these which arise in physiology. Here, and in the following three chapters, we shall consider some deterministic models. The book edited by May (1981) gives an overview of theoretical ecology from a variety of different aspects; experts in diverse fields review their areas. The book by Nisbet and Gurney (1982) is a comprehensive account of mathematical modelling in population dynamics: a good elementary introduction is given in the textbook by Edelstein-Keshet (1988).
1.1 Continuous Growth Models Single species models are of relevance to laboratory studies in particular but, in the real world, can reflect a telescoping of effects which influence the population dynamics. Let N(t) be the population of the species at time t, then the rate of change dN = b'lrths - deat hs + mIgratIOn, . . dt (1.1) is a conservation equation for the population. The form of the various terms on the right hand side of (1.1) necessitates modelling the situation that we are concerned with. The simplest model has no migration and the birth and death terms are proportional to N. That is
dN =bN-dN dt
N(t)
= No e(b-d)t
where b, d are positive constants and the initial population N(O) = No. Thus if b > d the population grows exponentially while if b < d it dies out. This J. D. Murray, Mathematical Biology © Springer-Verlag Berlin Heidelberg 1993
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1. Continuous Population Models for Single Species
approach, due to Malthus in 1798 but actually suggested earlier by Euler, is pretty unrealistic. However if we consider the past and predicted growth estimates for the total world population from the 17th to 21st centuries it is perhaps less unrealistic as seen in the following table.
Date
Population in billions
Mid 17th Century
Early 19th Century
1918-27
1960
1974
1987
1999
2010
2022
0.5
1
2
3
4
5
6
7
8
In the long run of course there must be some adjustment to such exponential growth. Verhulst in 1836 proposed that a self-limiting process should operate when a population becomes too large. He suggested
dN
dt
= rN(l - NIK) ,
(1.2)
where rand K are positive constants. This is called logistic growth in a population. In this model the per capita birth rate is r(l - N I K), that is, it is dependent on N. The constant K is the carrying capacity of the environment, which is usually determined by the available sustaining resources. There are two steady states or equilibr
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