It makes sense to consider species that do not reproduce continuously.  Many fish and birds, for example, only reproduce seasonally.  Many insects reproduce at a specific stage in their lifestyle.  We consider these discrete reproducers.

Presume that each adult will produce, on average,  β new adults by the next season, and that each adult has a γ chance of dying before the next season. (β  is not the birth rate, but the number of offspring that
survive to adult-hood).   Therefore, the population in the next year, Nn+1, will be the population this year (Nn) plus the number of new adults to survive, minus the number of existing adults to die.

Nn+1 = Nn + βNn −  γNn = (1 +  β −  γ)Nn = RoNn

While the growth here is exponential, the growth happens in discrete stages.  The future population is being described by the current one is called the difference equation.

The model is graphed:

Each Nn feeds into the next generation, so the Nn+1 becomes the Nn of the next time point (this is why the graph ‘bounces’ off the x=y slope).  If graphed over time, the population would increase exponentially, but not as a smooth curve but a series of ascending plateaus.

But, of course, natural populations do not experience this type of growth.  We can use this type of model to observe different types of intraspecies competition.  There are two ends of the competition spectrum.  Either a winner take all system, where some in the population can outcompete others, and so the population stabilizes to  some maximum level (called contest competition).  Or a ’share the pie’ type, where all individuals are equally good at competing, and share a smaller slice of the pie as new individuals are added.  This population (in scramble competition) crashes when the environment can longer sustain the population.

Graphed, these two types look like:

Scramble

Where Nn+1 = RoNn if  Nn< Nc but Nn+1 = 0 if Nn > Nc

contest

Where Nn+1 = RoNn if  Nn< Nc but Nn+1 = RoNc if Nn > Nc

In natural populations, however, you’ll rarely find a population that can be described completely by either contest or scramble competition (or complete stabilization or complete collapse) so it would be useful if we had an equation that could describe ‘in between’ states.

The Hassel Equation is as follows:

; where a & b are positive constants.

for b = 0, the equation becomes like the difference equation and for b = 1, we get a model of contest competition (figure a) below).  For b > 1, we get overcompensation and oscillation around the stable maximum (figur b) below).  As b increases, the population becomes more unstable and crashes almost to zero (figure c) below)

Next time, we continue our analysis in interspecies competition!

This entry was posted on Monday, March 16th, 2009 at 9:57 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.