Experiments in Minimal Food Web Evolution
dX/dt = R1 * X – B1 * X * Y
dY/dt = -R2 * Y + B2 * X * Y
Lotka-Volterra is simplified to two parameters.
B1 is the consumption coefficient. B2 is 1/10 * B1 (by the Ecological Constant of trophic interactions)
R1 is the growth coefficient. R2 is 2/3 * R1(it’s just convenient)
Population levels are set at the stable equilibrium. Stable equilibrium is at (R2/B2, R1/B1)
A single individual is removed from the predators, mutated by modifying the consumption coefficient, and
put in a second predator population.
Extinction occurs when the population dips below 0.1.
How does Lotka-Volterra react to single mutation in predator?
The mutant predator is established only if the consumption coefficient is increased. Any mutations, to any
of the coefficients, bumps the system to a new L-V state space once the original predator goes extinct.
The new state space is most likely disturbed from a stable equilibrium. If the change in state space is
thought of as a multidimensional vector, then the direction and magnitude of that vector (defined by the
mutation level and original coefficients) determines the amount of state space disturbance.
Example Mutant takeover. Consumption is 0.6, Growth is 0.05, Mutation is 1.1.
How long does it take to for original predator to go extinct?
Here the mutation level is varied, while the coefficients are held constant. The time to extinction decays
exponentially with the amount of mutation.
Here the consumption coefficient is varied, while mutation and growth is held constant. Very small
consumption coefficients lead to extinction of all predators.
Here the growth coefficient is varied, while mutation and consumption is held constant.
How disturbed is the L-V cycle?
State Disturbance is measured by taking the ratio of population variation to population mean.
Generally, the greater the mutation is, the greater the disturbance. Any increase in consumption
coefficient will decrease the stable population point.
The amount of disturbance is crucially dependent on when extinction happens in the L-V cycle. Here
mutation level is varied.
Varying the Growth Coefficient leads to increased disturbance.
Increasing consumption leads to decreased disturbance.
What happens to mutations in L-V already away from equilibrium?
Only when both Predator and Prey populations are increasing (1/4 of cycle) can mutant become
established.
Mutant bumps the state space a bit, then goes extinct
Mutant established only ¼ of the 14 step cycle.
So, difficulties in constructing food webs from simple beginnings are understandable. With the addition of
another prey species before predator mutation, establishment might be easier and coexistence possible.
That’s a next step, and then studying single controlled mutations in more complex stable food webs for
cascades of state shifts.