## The Mathematics Behind Biological Invasions by Mark A. Lewis, Sergei V. Petrovskii, Jonathan R. Potts

By Mark A. Lewis, Sergei V. Petrovskii, Jonathan R. Potts

This e-book investigates the mathematical research of organic invasions. not like in simple terms qualitative remedies of ecology, it attracts on mathematical thought and techniques, equipping the reader with sharp instruments and rigorous technique. topics contain invasion dynamics, species interactions, inhabitants unfold, long-distance dispersal, stochastic results, chance research, and optimum responses to invaders. whereas in accordance with the idea of dynamical platforms, together with partial differential equations and integrodifference equations, the ebook additionally attracts on info thought, computer studying, Monte Carlo tools, optimum regulate, records, and stochastic approaches. purposes to genuine organic invasions are incorporated all through. eventually, the booklet imparts a strong precept: that by means of bringing ecology and arithmetic jointly, researchers can discover new knowing of, and potent reaction options to, organic invasions. it's compatible for graduate scholars and tested researchers in mathematical ecology.

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**Example text**

35). 26) without calculating the eigenvalues, using only the relative positions of the nullclines. u; v/ D 0. Then the following theorem holds. 35). u/. 26), we need to know the signs of the trace and determinant of the linearized system’s matrix. A/. 35). u/ is descending, dvP =du < 0 and thus Pu < 0. u/ is ascending, dvQ =du > 0 and thus Qv < 0. A/. Since Pv < 0 and Qu > 0, Pv Qu < 0. As is proven above, Pu < 0 and Qv < 0, so that Pu Qv > 0. 45) that the steady state is stable. Note that this theorem gives a sufficient condition for stability but not a necessary one.

4, the populationdynamics models originally designed for species spread can also provide insight into some peculiarities of species establishment, including a possible mechanism for the time lag. 2 Population Dynamics in a Nonspatial System We have shown in the previous sections that biological invasion is a complicated phenomenon that results from the interactions between various processes and factors: population biology, genetics, environmental biochemistry, economics, social science, etc. A comprehensive understanding of the invasion dynamics, as well as of the impact of the alien species on ecology, the environment, and society, apparently requires these numerous factors to be taken into account in an interdisciplinary theoretical framework.

Number of papers on biological invasions The examples in Sect. 1 show that biological invasion is an old problem. However, its systematic study did not begin until the middle of the twentieth century. The birth of invasion ecology as a science is often linked to Elton’s book [95], which was published in 1958 (see also [326]), although ideas fully relevant to alien species spread had been considered by Fisher some 20 years earlier [114]. There are a large number of scientific publications addressing various aspects of biological invasion.