MIMIC - Example

Example

Problem

Consider a predator-prey model from the field of marine biology to determine the dynamics of fish and shark populations. As a simple model, we choose the Lotka–Volterra equation and the constants given in a tutorial.

If

f(t): Fish population over time (fish)

s(t): Shark population over time (sharks)

df / dt or : growth rate of fish population (fish/year)

ds / dt or : growth rate of shark population (sharks/year)

: growth rate of fish in the absence of sharks (1/year)

: death rate per encounter of fish with sharks (1/sharks and year).

: death rate of sharks in the absence of their prey, fish (1/year)

: efficiency of turning predated fish into sharks (sharks/fish)

then

wlth initial conditions

The problem's constants are given as:

• = 600 fish
• = 50 sharks
• = 0.7 fish/year
• = 0.007 fish/shark and year
• = 0.5 shark/year
• = 0.1 shark/fish
• tmax = 50 year

Code sample

Card columns 0 1 2 3 4 5 6 7 12345678901234567890123456789012345678901234567890123456789012345678901 ----------------------------------------------------------------------- * A SIMPLE PREDATOR-PREY MODEL FROM MARINE BIOLOGY / (TUTORIAL 2: NUMERICAL SOLUTION OF ODE'S - 19/08/02) / ENVIRONMENTAL FLUID MECHANICS LAB / DEPT OF CIVIL AND ENVIRONMENTAL ENGINEERİNG / STANFORD UNIVERSITY * * LOTKA–VOLTERRA EQUATION CON(F0,S0,TMAX) CON(ALPHA,BETA,GAMMA,EPS) 1DF = ALPHA*F-BETA*F*S F = INT(1DF,F0) 1DS = EPS*BETA*F*S-GAMMA*S S = INT(1DS,S0) HDR(TIME,FISH,SHARK) OUT(T,F,S) PLO(F,S) FIN(T,TMAX) END 600. 50. 50. 0.7 0.007 0.5 0.1