Hutchinson equation

Suppose a population inhabits a bounded homogeneous area $ \Omega \subset \mathbf R ^ {2} $ with piecewise-smooth boundary $ \partial \Omega $. 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 [a1] it was postulated that under these idealized conditions the variation in population density $ n ( t ) = { {N ( t ) } / K } $, where $ N ( t ) $ is the current size of the population and $ K $ is the average number of the population, which depends upon the size of the habitat and the amount of food available, obeys the law

$$ {\dot{n} } = \lambda [ 1 - n ( t - 1 ) ] n, \quad \lambda > 0. $$

Here, $ \lambda $ 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 $ \lambda < {\pi / 2 } $, the attractor of positive solutions of the equation is its unit state of equilibrium, whilst for $ \lambda > {\pi / 2 } $ 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, [a2]).

As $ \lambda $ increases, the Hutchinson cycle $ n _ {0} ( t, \lambda ) $ acquires a distinctive relaxation character, which is evident from the following facts [a3]. Assume, for the sake of being specific, that $ n _ {0} ( - 1, \lambda ) = 1 $ and $ {\dot{n} } _ {0} ( - 1, \lambda ) > 0 $. Then the largest value $ n _ { \max ,0 } $ of the function $ n _ {0} ( t, \lambda ) $ is reached at $ t = 0 $:

$$ n _ { \max ,0 } = { \mathop{\rm exp} } ( \lambda - 1 ) + { \frac{1}{2e } } + O ( { \mathop{\rm exp} } ( - \lambda ) ) . $$

The smallest value, $ n _ { \min ,0 } $ of the function $ n _ {0} ( \lambda,t ) $ is realized at $ t = 1 + t _ {0} $, where

$$ t _ {0} = { \frac{ { \mathop{\rm ln} } \lambda }{\lambda - 1 } } + O \left ( { \frac{ { \mathop{\rm ln} } ^ {2} \lambda }{\lambda ^ {3} } } \right ) . $$

The asymptotic equality

$$ { \mathop{\rm ln} } n _ { \min ,0 } = - { \mathop{\rm exp} } \lambda + 2 \lambda - 1 + { \frac{1 + ( 1 + \lambda ) { \mathop{\rm ln} } \lambda } \lambda } + $$

$$ + O \left ( { \frac{ { \mathop{\rm ln} } ^ {2} \lambda }{\lambda ^ {2} } } \right ) $$

is valid.

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

$$ {\dot{n} } = D \Delta n + \lambda [ 1 - n ( t - 1,x ) ] n, \quad \left . { \frac{\partial n }{\partial \nu } } \right | _ {\partial \Omega } = 0, $$

where $ x \in \mathbf R ^ {2} $, $ \Delta $ is the Laplace operator, $ D $ is the mobility coefficient, and $ \nu $ is the direction of the external normal. When $ D $ 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 [a3].

References

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