By Haccou P., Jagers P., Vatutin V.A.
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"This paintings develops the technique in keeping with which periods of discontinuous capabilities are utilized in order to enquire a correctness of boundary-value and preliminary boundary-value difficulties for the instances with elliptic, parabolic, pseudoparabolic, hyperbolic, and pseudohyperbolic equations and with elasticity thought equation structures that experience nonsmooth strategies, together with discontinuous options.
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If reproduction or survival are age-dependent, multi-type processes must be used throughout. If no other factors are involved, type corresponds to age. 6 As a further illustration, consider a biennial plant species. If it were strictly biennial, 1-year-old individuals produce no seeds and have a positive chance p1 of survival, but 2 year olds would always produce seeds and die. Since such plant species usually have enormous amounts of seeds, only a few of which germinate, a Poisson offspring distribution seems natural.
We consider the situation in which it is possible to choose left eigenvectors u so that their components are nonnegative and sum to one. Then the corresponding right eigenvectors v also have non-negative entries and can be chosen so that u T v = 1. In explicit terms, d uT v = d u j v j = 1, v j ≥ 0, u j ≥ 0, j =1 uj = 1 . 32) 26 Branching Processes: Variation, Growth, and Extinction of Populations We now use this normalization of the eigenvectors. 30). Thus, the eigenvalue ρ is the required indicator of criticality in the direction of the eigenvector: the process is subcritical if ρ < 1, critical if ρ = 1, and supercritical if ρ > 1.
Populations with m < 1, m = 1, or m > 1 are called subcritical, critical, and supercritical, respectively. 12), we see that as n → ∞ the average size of a subcritical population goes to zero and that of a critical population is stable, whereas the average size of a supercritical population grows unboundedly. Indeed, both these developments occur at the famous exponential rate: there is decay at the rate m n if m < 1, and growth at the rate m n if m > 1. Exactly critical populations, however, are of less interest for applications, since it is not reasonable to assume that the average number of direct descendants per individual is exactly one during many generations.
Branching processes by Haccou P., Jagers P., Vatutin V.A.