By Mimmo Iannelli, Andrea Pugliese (auth.)
This e-book is an creation to mathematical biology for college kids without adventure in biology, yet who've a few mathematical history. The paintings is targeted on inhabitants dynamics and ecology, following a convention that is going again to Lotka and Volterra, and encompasses a half dedicated to the unfold of infectious illnesses, a box the place mathematical modeling is very renowned. those issues are used because the region the place to appreciate kinds of mathematical modeling and the prospective that means of qualitative contract of modeling with facts. The booklet additionally contains a collections of difficulties designed to process extra complex questions. This fabric has been utilized in the classes on the college of Trento, directed at scholars of their fourth yr of reports in arithmetic. it could actually even be used as a reference because it presents up to date advancements in different areas.
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Additional info for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka
31) K In this case, the growth rate has the form r(N) = r 1 − N K − aP∗ , and the term aP∗ (predation rate) is an extra mortality term giving the fraction of individuals dying in the unit of time, due to the predation mechanisms. 31) we study the action of a generalist predator on a prey due to predator attacks subjected to a logistic effect. By the transformation t → t˜ = rt, u= N , K we rescale the model into u (t) = F(u(t)) = (1 − u(t))u(t) − ρ u(t), ∗ where ρ = aPr is a dimensionless parameter, measuring the action of the predator.
1/(n1 n3 ) − 1/n22 Prove this formula and use it in the previous point. 2. 10) with periodic Malthus parameter: r(t + T ) = r(t) and take the average over one period r¯ = 1 T T r(s)ds. 0 1. 13). 39) 34 1 Malthus, Verhulst and all that 2. 40) assuming both r(t) and m(t) periodic with period T. 40) and is periodic with period T. 3. e. lim (N(t) − N∞ (t)) = 0. 3. Intraspeciﬁc competition Assume that the growth rate of a population depends on available resources ρ (t), according to a general law r(t) = G(ρ (t)).
Thus we consider a single population living isolated, in an invariant habitat, all of its individuals being perfectly equal but for their age. In accordance with this phenomenological setting, fertility and mortality are intrinsic parameters of the population and do not depend on time, nor on the population size: they are functions of age only. Thus we introduce: • β (a): age-speciﬁc fertility, the mean number of newborns per unit time, borne by each individual whose age is in the inﬁnitesimal age interval [a, a + da], at time t; • μ (a): age-speciﬁc mortality, the death rate of individuals having age in [a, a+ da] at time t.
An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka by Mimmo Iannelli, Andrea Pugliese (auth.)