# A predator-prey system

From 1909 through 1932, the Hudson Bay Trading Company reported the following yearly numbers of hare and lynx pelts sold:

1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932
Hare 25 50 55 75 70 55 30 20 15 15 20 35 60 80 85 60 30 20 10 5 5 10 30 80
Lynx 2 4 10 14 19 14 8 9 2 1 1 2 4 4 8 7 9 7 4 3 2 3 3 5

When plotted against time, like in the graph to the right, the populations of hare and lynx seem to oscillate in a common rhythm. It was suggested that lynx feed on hare and reproduce well if they find enough prey. However, if their number grows their consumption of hare increases too thereby eventually worsening their living conditions (Odum 1994).

In the 1920ies, two scientists - Alfred Lotka (1913) and Vito Volterra (1926) – suggested to model this interrelation with coupled first-order differential equations. The following is a simplified mathematical abstraction of these equations that is commonly referred to as the Lotka-Volterra- or predator-prey-equations:

$\frac{dH}{dt}=H*(a-b*L)$ $\frac{dL}{dt}=-L*(c-d*H)$

saying that $$H$$, the hare population, grows in respect to its birthrate $$a$$ in the absence of predators, and it declines in respect to its death-rate per predator $$b$$ i.e. the strength of influence of the predator population of lynx $$L$$.

The predator population lynx $$L$$ shrinks in respect to its death rate $$c$$ in the absence of prey, and it grows in respect to its reproduction rate per prey $$d$$.

The following interactive model lets you test some of the reactions of this model to parameter changes.

The small inset in the right upper corner shows a state space or phase space portrait. It depicts the number of predator against the number of prey.

It should be noted that this model is more of a mathematical curiosity and has rather limited biological relevance. The reason is that any small change of the model will lead to a qualitatively different type of behavior. The model is therefore said to be “structurally unstable”. Mathematicians use it in teaching examples because the model is so elegantly simple.

Lotka, A. J. (1913). A natural population norm. J. Wash. Acad. Sci. 3, 289–293.

Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560.

Odum, Howard (1994). Ecological and General Systems: An introduction to systems ecology, Colorado University Press, Colorado.

Implemented with a System Dynamics software like Vensim this model might look like follows: