By Mimmo Iannelli, Andrea Pugliese (auth.)
This e-book is an advent to mathematical biology for college kids with out adventure in biology, yet who've a few mathematical history. The paintings is concentrated on inhabitants dynamics and ecology, following a convention that is going again to Lotka and Volterra, and contains a half dedicated to the unfold of infectious ailments, a box the place mathematical modeling is very well known. those subject matters are used because the region the place to appreciate varieties of mathematical modeling and the prospective which means of qualitative contract of modeling with facts. The ebook additionally contains 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 could even be used as a reference because it presents up to date advancements in numerous areas.
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Extra info for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka
31) and of the spruce-budworm model are good examples of the concepts introduced above. 9 The spruce-budworm system The spruce-budworm dynamics concerns the ecological case study of the interaction between the spruce-budworm and the balsam-ﬁr forests in Eastern North America. The spruce-budworm is in fact one of the insect pests affecting vegetation, through periodic outbreak that represent a great concern for forest conservation. Apart from extensive qualitative studies, the spruce-budworm has been the object of a detailed modeling effort, in order to identify mechanisms apt to explain the dynamics.
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.
More technical computations are left to a ﬁnal section. 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.
An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka by Mimmo Iannelli, Andrea Pugliese (auth.)