By Mimmo Iannelli, Andrea Pugliese (auth.)
This e-book is an creation to mathematical biology for college kids with out adventure in biology, yet who've a few mathematical historical past. The paintings is targeted on inhabitants dynamics and ecology, following a practice that is going again to Lotka and Volterra, and encompasses a half dedicated to the unfold of infectious ailments, a box the place mathematical modeling is very renowned. those issues are used because the sector the place to appreciate types of mathematical modeling and the prospective that means of qualitative contract of modeling with information. The publication 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 yr of stories in arithmetic. it might probably even be used as a reference because it presents updated advancements in numerous areas.
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Extra resources for An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka
Economic theory predicts that, for an open access ﬁshery, the effort E will in the long run reach the value at which the gain is equal to 0. e, ﬁnd, if they are greater or smaller) to those in the previous case. 3. Let assume that the government taxes at a percentage θ the gains obtained by ﬁsheries. How does this affect the results obtained with open-access ﬁshery? 4. Consider again open-access ﬁshery and assume that the government taxes according to how much has been ﬁshed Y . Let us consider two separate cases: a constant fraction θ Y , or a progressive tax τ (Y ) given by the formula τ (Y ) = 0 if Y ≤ Y0 θ (Y −Y0 ) if Y > Y0 Which are the results of these regulations?
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.
18) has the two equilibrium points (see Fig. 4) E0 = (0, 0), E ∗ = (1, 1) . We study their stability by looking at the eigenvalues of the Jacobian matrix at each of them (see Appendix A) Concerning E0 the Jacobian matrix is ⎛ 1 J(E0 ) = ⎝ 1 τ 0 − ⎞ 1 ⎠, τ 48 2 Population models with delays with eigenvalues λ1 = 1 and λ2 = −1/τ . Thus E0 is a saddle point (unstable). The equilibrium E ∗ is instead stable since ⎛ 0 −1 J(E ∗ ) = ⎝ 1 τ − ⎞ 1 ⎠, τ and 1 Trace (J(E ∗ )) = − < 0, τ det(J(E ∗ )) = 1 > 0.