By Mimmo Iannelli, Andrea Pugliese
This publication is an advent to mathematical biology for college students without adventure in biology, yet who've a few mathematical historical past. The paintings is concentrated on inhabitants dynamics and ecology, following a convention that is going again to Lotka and Volterra, and incorporates a half dedicated to the unfold of infectious ailments, a box the place mathematical modeling is intensely well known. those subject matters are used because the zone the place to appreciate forms of mathematical modeling and the prospective that means of qualitative contract of modeling with info. The booklet additionally encompasses a collections of difficulties designed to strategy extra complex questions. This fabric has been utilized in the classes on the college of Trento, directed at scholars of their fourth 12 months of reviews in arithmetic. it may well even be used as a reference because it presents updated advancements in different components.
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Extra resources for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka (UNITEXT, Volume 79)
9 The spruce-budworm system (a) 29 (b) Fig. 36 in three different cases (a), (b) and (c) respectively corresponding to ρ < ρ1 , ρ ∈ (ρ1 , ρ2 ) and ρ > ρ2 From the same Fig. 16 we can determine the stability of the states shown in Fig. 17. In fact, in Fig. 17a we identify three branches of steady states u∗1 (ρ ) for ρ ∈ [0, ρ2 ], u∗2 (ρ ) for ρ ∈ (ρ1 , ρ2 ), u∗3 (ρ ) for ρ ∈ [ρ1 , +∞) satisfying 0 < u∗i (ρ ) < K, i = 1, 2, 3, u∗1 (ρ ) < u∗2 (ρ ) < u∗3 (ρ ), for ρ ∈ (ρ1 , ρ2 ) . Remember that F(u) = r(u) = ( f (u) − g(u)) u, so that F (u∗i (ρ )) = r (u∗i (ρ ))u∗i (ρ ) .
E. the fertility at low population size. 28) r(N) = β e−ν N − μ (scramble competition). 29) r(N) = and Both cases correspond to a constant mortality μ , and respective fertility β (N) = and βK K +N (with r = β − μ ) β (N) = β e−ν N . 8 Generalist predation Among the many factors inﬂuencing the growth of a species, we should include the interaction with other species; in particular, predators represent an additional source of mortality because they produce a direct loss of individuals. Actually, the mechanisms regulating predator-prey interaction should include a feedback effect from prey abundance on predator growth, so that a suitable framework would be a two species model.
1 Italo Calvino, “Gli anni luce” in “Le Cosmicomiche”, 1965 The possibility that events, lost in the past, still inﬂuence our lives, is an evocative notion, strongly inspiring literature, at least. The character of the tale by Italo Calvino, quoted in the foreword, is in fact obsessed by some event happened millions of years before, within an hallucinatory mixture of space and time. However, the need of including the inﬂuence of past effects in the description of natural phenomena was explicitly recognized by Vito Volterra in a lecture in 1912 ( ), where we read 2 : at this point we may ask if natural phenomena really occur in such a way, or else we have to suppose that there effectively exists an heredity of past events, so that disregarding such heredity represents an approximation introduced only for study convenience.
An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka (UNITEXT, Volume 79) by Mimmo Iannelli, Andrea Pugliese