# An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka (UNITEXT)

## Mimmo Iannelli, Andrea Pugliese

Language: English

Pages: 346

ISBN: 3319030256

Format: PDF / Kindle (mobi) / ePub

This book is an introduction to mathematical biology for students with no experience in biology, but who have some mathematical background. The work is focused on population dynamics and ecology, following a tradition that goes back to Lotka and Volterra, and includes a part devoted to the spread of infectious diseases, a field where mathematical modeling is extremely popular. These themes are used as the area where to understand different types of mathematical modeling and the possible meaning of qualitative agreement of modeling with data. The book also includes a collections of problems designed to approach more advanced questions. This material has been used in the courses at the University of Trento, directed at students in their fourth year of studies in Mathematics. It can also be used as a reference as it provides up-to-date developments in several areas.

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order to deﬁne a global model, we have to take into account also the logistic effect previously discussed, so that we end up in the following constitutive assumption for the growth rate r(N): there exists Nm such that i) r (N) > 0 ii) r(Nm ) > 0, for N < Nm , r (N) < 0 for N > Nm , (1.23) lim r(N) < 0. N→∞ Condition (1.23) implies that there exists at least an asymptotically stable equilibrium at N = K > Nm and possible forms of r(N) are shown in Fig. 1.11. Figure 1.11a shows the case r(0) >

computer. We see nothing wrong with that kind of models, but our basic choice here is to emphasize simple models that generically (i.e. over a reasonable range of parameter values) yield patterns in qualitative agreement with characteristic features of the biological phenomenon under study. In fact, in our view, understanding what are the consequences of model structure and assumptions in simple cases is essential also when one attempts to build a complex model, and to understand the result of

response Holling II. 6. Repeat the study of the previous item with a functional response Holling III. 7. Consider the case of a prey with strong Allee effect, modeled by (1.25), and a generalist predator with the three functional responses considered in the previous items. 36 1 Malthus, Verhulst and all that 1.6. Harvesting problems Let us consider a population growing according to a logistic dynamics. Let assume that a constant effort E of ﬁshing3 is exerted so that the yield per unit time

the carrying capacity K, hence, if we increase the supply of energy to an ecosystem we would not produce any increase of the prey abundance, but only give an advantage for the predator to grow. 166 6 Predator-prey models Fig. 6.15 The chemostat: a container with a ﬂuid continuously stirred to guarantee homogeneous mixing, and a continuous input and output of ﬂuid in order to convey nutrient and renew the environment of the culture 6.5 Growing bacteria in a chemostat The ideas developed in

1.2) r∗ = r = 1 T T r(s)ds. (1.12) 0 Moreover, in this case, one has a result more precise than (1.11), since we have N(t) = er t Nπ (t), (1.13) where Nπ (t) is a periodic function. Thus the growth is an exponential tuned by oscillations (see Fig. 1.5). 1.4 Endogenous variability of the habitat Concerning endogenous variability of the habitat we have to modify Malthus model to include all possible factors that inﬂuence species growth, through the many possible effects that the presence