By Mimmo Iannelli, Andrea Pugliese
This booklet is an advent to mathematical biology for college kids with out adventure in biology, yet who've a few mathematical heritage. The paintings is targeted on inhabitants dynamics and ecology, following a practice 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 issues are used because the sector the place to appreciate sorts of mathematical modeling and the prospective which means of qualitative contract of modeling with information. The booklet additionally encompasses a collections of difficulties designed to procedure extra complicated questions. This fabric has been utilized in the classes on the collage of Trento, directed at scholars of their fourth 12 months of stories in arithmetic. it may even be used as a reference because it offers up to date advancements in different parts.
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Extra resources for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka (UNITEXT, Volume 79)
1 Why delays, and how When trying to describe the status of an ecosystem at a certain time, we should be aware of the fact that some of the mechanisms involved in the dynamics may depend on the status of the population at past times, more than at present. First, we note that reproduction must reﬂect the status of the population at some moment in the past, since gestation times imply that present birth rates must depend on the number of individuals at fecundation more than those at present. A more general reason for considering delays is that some of the habitat features inﬂuencing population growth at a given time may be inﬂuenced by the population abundance in the past.
21) we have, B(t) = a† 0 β (a)Π (a)B(t − a)da, t ≥ 0. 24) Cumulative variables Though the age distribution n(a,t) contains all informations about the population state, other variables combining n(a,t) with age-speciﬁc parameters and rates may be signiﬁcant. 25) which, in the same spirit, gives the total number of deaths per unit time. 27) where a∗ denotes a “maturation” age. More generally, we may be interested in weighted selections such as a† S(t) = 0 γ (a)n(a,t)da. 28) All these cumulative variables are important for models design and development.
15) have negative real part. 15) are in the left hand side of the complex plane. 15) changes with τ . From the analysis of the change in the location of roots, reported at the end of this chapter in Sect. 6 and displayed graphically in Fig. 15) cross the imaginary axis so that the equilibrium solution u∗ = 1 becomes unstable. 46 2 Population models with delays Fig. 15) in the complex plane as τ varies. At τ = when two complex roots cross the imaginary axis π 2 Hopf bifurcation occurs This result is a prelude to Hopf bifurcation as discussed in Appendix A; numerical evidence shows that Hopf bifurcation actually occurs.
An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka (UNITEXT, Volume 79) by Mimmo Iannelli, Andrea Pugliese