One approach to mathematical modeling is to observe the system and note the relative frequency with which various events occur. One can then mimic this behavior by randomly sampling these occurrences in a computation at the observed relative frequency. This class of modeling techniques is generally known as Monte Carlo Method.
The physical system that will be modeled is the evolution of two populations, predators and preys, within a segmented spatial extent. The predator population is reduced in the absence of prey. (A predator that has not eaten for ten cycles, days, will starve.) The predator will multiply (slowly) in the presence of prey. The prey population is diminished by the predators and multiplies proportionally to the prey population. (This physical system is also exemplified in chemistry by reaction rate equation and, in general, in physics for competing growth constraints.)
The predator/prey system has been described in several treatises. Lotka and Volterra [ref.] independently produced parameterized closed form differential equations for the evolution of the system (without regard to spatial extent). Their equations:
R(n+1) = R(n) + a*R(n) - b*R(n)*F(n)
F(n+1) = F(n) + e*b*R(n)*F(n) - c*F(n)
where the parameters are defined by:
• a is the natural growth rate of rabbits in the absence of predation,
• c is the natural death rate of foxes in the absence of food (rabbits),
• b is the death rate per encounter of rabbits due to predation,
• e is the efficiency of turning predated rabbits into foxes.
Can be analyzed to show that stable, oscillating solutions exist.
I hope expand on such work to include migration. Migration, in this model, occurs isotropically. The probability that a prey will change direction, isotropically, is 0.3 [per cycle]. The prey’s speed will be 10* cell’s prey population/total prey population. In cells with high densities of prey, the prey will move faster so as to redistribute the prey population (biologically, this might be the result of the competition for food.) The probability that a predator will change direction, isotropically, is 0.3 [per cycle]. The predator’s speed will be 0.1*(1.05^days since last meal.), (In biology this correlate with the increasing need to range further to find food the hungrier the predator becomes.)
It is well known that Monte Carlo simulations converge slowly to a description of the physical system’s state. That is, it takes a large number of event simulations to adequately sample the possible evolutionary paths. In particular, quadrupling the number of simulated events only halves the error in the approximation to the solution [ref: central limit theorem]. Nevertheless, the extent to which the event simulations are independent of another allows the power of massively parallel computation to compensate for this inherent inefficiency. This project intends implement a Monte Carlo algorithm on a parallel computer to explore the viability of this approach to mathematical modeling.
To date, I have studied Monte Carlo Method, reviewed computer programming, and begun on the program.
I hope to find stable solutions within this model.
References:
Lokta Volterra Model from:
http://www.stolaf.edu/people/mckelvey/envision.dir/nonDE.lotka-volt.html
Monte Carlo Method:
Carter, Cashwell. Particle-Transport Simulation with the Monte Carlo Method.
C Programming Review:
Roberts. The Art and Science of C.
Other Works Using Predator Prey Models and Migration:
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11822547&dopt=Abstract
http://www.ovcrd.upd.edu.ph/vol%2011%20no1%20abs%203.htm
And I would like to thank my mentor, Steve White, for all his assitance.