Predator vs Prey - Will the Wolves Dominate?

An occasional excursion from topics photographic into the realm of mathematics and statistical modeling.  I will cover some modeling mathematics for the interactions of predatory and prey in a series of three posts and include a program for the reader to use for further exploration.
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Introduce a herd of deer into a forest.  Let the population grow.  Now add a predator, like a family of wolves.  What will happen?  It depends.  A modeling program can be useful to explore the possibilities.

According to some basic principles of ecology, populations grow following some fairly simple mathematics.  Thomas Malthus (http://en.wikipedia.org/wiki/Thomas_Robert_Malthus) was familiar with these principles and the basics have not changed much since then.  A given population of mice, squirrels, deer, or humans has an intrinsic rate of growth that is limited by the carrying capacity of the environment.

Stating this in mathematical terms:

Population at time T+1 = Population at time T + Growth between T and T+1
Growth = intrinsic rate of growth x carrying capacity of the environment

Of course the carrying capacity is not a constant.  For example, advances in food production technology can increase the food availability for growth of human populations.  Conversely, climatic variation could result in vegetative variation causing swings in the carrying capacity for a theoretical population of deer.

Shown in the first chart is the plot of a deer herd population that grows under ideal conditions: the carrying capacity is reasonably fixed and there are no predators to eat the deer.  They live a happy, idilic life.  At the start there is a herd of 5000 deer growing at a rate of 50%/year (they are VERY randy deer).  The carrying capacity (blue line) is 20000 and the relative variability in the carrying capacity is less than 1%.  The deer herd (green line and green dots) grows rapidly until it reaches the carrying capacity and stabilizes without further variation.  The wolf family (red) has not been introduced yet so the red line is flat with a constant value of zero.


Chart 1: Constant carrying capacity and no predators

Of course, if the environment is not constant, neither will the carrying capacity be constant.  If variation is added to the equation for carrying capacity, the deer population will also fluctuate as in the next chart where the carrying capacity has been assigned a variability of 40%.  The variation in the deer herd (green) matches the variation in the carrying capacity (blue).  Over the short term, the deer herd survives, with a population size that sometimes exceeds and other times is lower than the average carrying capacity. 

Chart 2: Variable carrying capacity and no predators

When the carrying capacity variation is increased to 50%, the deer herd still shows rapid growth initially from a starting value of 5000 until it reaches the carrying capacity of 20000, then drops rapidly to 10000, rebounds, but then crashes to zero.  Of course, once the herd size reaches zero, it is difficult to rebound and the population remains at zero after just 20 generations.  

Chart 3: Highly variable carrying capacity and no predators

In the next posting I will introduce a family of wolves and see how this changes the dynamics of the herd's population growth.  In the last posting I will provide you with a programmed spreadsheet where you can experiment with different starting conditions and see for yourself how the environment, prey, and predator interact.