View source for Hutchinson equation

Suppose a population inhabits a bounded homogeneous area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103801.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103802.png" />. Assume that its food base regularly restores itself to a certain level, whilst the migration factor is so high that complete mixing takes place. In [[#References|[a1]]] it was postulated that under these idealized conditions the variation in population density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103803.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103804.png" /> is the current size of the population and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103805.png" /> is the average number of the population, which depends upon the size of the habitat and the amount of food available, obeys the law

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Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103807.png" /> is the Malthusian coefficient of linear growth, provided that the age of sexual maturity of females is taken as the unit of time. This equation is called the Hutchinson equation. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103808.png" />, the attractor of positive solutions of the equation is its unit state of equilibrium, whilst for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h1103809.png" /> it is an orbitally exponentially stable cycle. This assertion is mainly based on the results of numerical analysis (significantly less was obtained by purely mathematical methods, [[#References|[a2]]]).

As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038010.png" /> increases, the Hutchinson cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038011.png" /> acquires a distinctive relaxation character, which is evident from the following facts [[#References|[a3]]]. Assume, for the sake of being specific, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038013.png" />. Then the largest value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038014.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038015.png" /> is reached at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038016.png" />:

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The smallest value, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038018.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038019.png" /> is realized at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038020.png" />, where

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The asymptotic equality

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is valid.

If migration across the habitat's boundary is forbidden, the migration factor results in the boundary value problem

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038026.png" /> is the [[Laplace operator|Laplace operator]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038027.png" /> is the mobility coefficient, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038028.png" /> is the direction of the external normal. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110380/h11038029.png" /> decreases, Hutchinson's cycle loses stability as a result of spacial perturbations connected with the appearance of so-called self-organization regimes, which are simultaneously complexly and regularly arranged towards spacial and temporal variables [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Hutchinson,   "Circular causal systems in ecology"  ''Ann. N.Y. Acad. Sci.'' , '''50'''  (1948–1950)  pp. 221–246</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Hale,   "Theory of functional differential equations" , Springer  (1977)  (Edition: Second)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.Yu. Kolesov,   Yu.S. Kolesov,   "Relaxation oscillation in mathematical models of ecology"  ''Proc. Steklov Inst. Math.'' , '''199''' :  1  (1995)  (In Russian)</TD></TR></table>