Hassell equation

The Hassell equation (also Hassel equation ) is one of several possible equations that are used in theoretical biology to describe density-regulated growth processes. Density regulation happens, for example, that a certain resource becomes scarce when the population grows, and thus the rate of growth decreases. The equation suggested by Michael Hassell has the form

${\displaystyle f_{r,b}(x)={\frac {rx}{(1+x)^{b}}}}$.

${\displaystyle x}$is the size or density of the population. The parameter describes the growth rate without density regulation, whereby with unlimited resources a constant population would mean an increase and a decrease. is a measure of the strength of the density regulation. ${\displaystyle r}$ ${\displaystyle r=1}$ ${\displaystyle r>1}$ ${\displaystyle r<1}$ ${\displaystyle b}$

For the term describes a so-called scramble competition , in which the scarce resource is divided equally among the individuals. This reduces the reproductive success for all individuals and the growth rate for large ones falls to zero. For the term describes a contest competition in which the scarce resource is divided unequally among the individuals, for example because some individuals can prevail against others. ${\displaystyle b>1}$ ${\displaystyle x}$ ${\displaystyle b<1}$

dynamics

For (population decreases despite sufficient resources), as clearly illustrated, only the trivial fixed point zero occurs independently of , which is also globally stable. ${\displaystyle r<1}$ ${\displaystyle b}$

For is obtained from the fixed point equation ${\displaystyle r>1}$

${\displaystyle x={\frac {rx}{(1+x)^{b}}}}$

another fixed point . ${\displaystyle x^{*}=r^{\frac {1}{b}}-1}$

For these two fixed points coincide. ${\displaystyle r=1}$

As known from the theory of dynamic systems , the sign and magnitude of the first derivative determine the stability of the fixed point and the behavior of the system in its environment. ${\displaystyle f'(x^{*},r,b)}$ ${\displaystyle x^{*}}$

It results:

${\displaystyle f'=1-b+{\frac {b}{r^{\frac {1}{b}}}}}$.

The zero crossing and thus the change from a stable system that oscillates around the fixed point to a stable system that converges monotonically towards the fixed point occurs at

${\displaystyle r=\left({\frac {b}{b-1}}\right)^{b}}$.

application

Substituting for a population density results in an expression which, under certain conditions, can be used to model the development over time of a population of organisms with non-overlapping generations. For example, the description of the temporal dynamics of insect populations is often done in this way. ${\displaystyle x}$ ${\displaystyle N}$

literature

• Nicholas F. Britton: Essential Mathematical Biology . Springer, ISBN 1-85233-536-X
• Introduction to Mathematical Biology - University of Stuttgart (PDF file; 2.40 MB)