Download An Introduction to Mathematical Population Dynamics: Along by Mimmo Iannelli, Andrea Pugliese (auth.) PDF

By Mimmo Iannelli, Andrea Pugliese (auth.)

This e-book is an advent to mathematical biology for college students with out event in biology, yet who've a few mathematical historical past. The paintings is concentrated on inhabitants dynamics and ecology, following a practice that is going again to Lotka and Volterra, and features a half dedicated to the unfold of infectious illnesses, a box the place mathematical modeling is intensely well known. those topics are used because the quarter the place to appreciate sorts of mathematical modeling and the prospective that means of qualitative contract of modeling with facts. The ebook additionally encompasses a collections of difficulties designed to strategy extra complicated questions. This fabric has been utilized in the classes on the college of Trento, directed at scholars of their fourth yr of reviews in arithmetic. it may even be used as a reference because it offers up to date advancements in numerous areas.

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Extra resources for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka

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1) is based on its asymptotic expansion over the set of exponential solutions of the equation. 24), we get the following characteristic equation for λ 1= ∞ 0 K(a)e−λ a da. 30) 54 2 Population models with delays This is called Lotka characteristic equation, where the right hand side is the Laplace transform of the kernel K(t). For this equation it is possible to obtain a result similar to that obtained for eq. 9). Namely we have: eq. 30) has one unique real solution λ0 = α ∗ . Any other solution λi is such that ℜλi < α ∗ and in each strip {λ ∈ C|ℜλ ∈ [a, b]} there is a finite number of roots.

5), we define Q(t) = 1 τ2 t −∞ (t − s)e− t−s τ u(s) ds, W (t) = 1 τ t −∞ e− t−s τ u(s) ds, and we get the three-dimensional system ⎧ u (t) = (1 − Q(t)) u(t), u(0) = u0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Q (t) = − 1 Q(t) + 1 W (t), Q(0) = − 1 0 se τs φ (s) ds, τ τ τ 2 −∞ ⎪ ⎪ ⎪ ⎪ 1 1 1 0 s ⎪ ⎪ ⎩ W (t) = u(t) − W (t), W (0) = e τ φ (s) ds. τ τ τ −∞ Here we similarly have the steady states E0 = (0, 0, 0), E ∗ = (1, 1, 1). 4); one can then easily see that it is a saddle point. Concerning E ∗ we have the Jacobian ⎞ ⎛ 0 −1 0 ⎟ ⎜ ⎜ 0 −1 1 ⎟ J(E ∗ ) = ⎜ τ τ ⎟ ⎟, ⎜ ⎝1 1⎠ 0 − τ τ 50 2 Population models with delays and its characteristic equation is τ 2 λ 3 + 2τλ 2 + λ + 1 = 0.

9 The spruce-budworm system 27 The procedure of scaling is however not univocal and it should be guided by the specific purposes of the analysis we wish to perform. 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.

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