Deterministic population dynamics · 5
Consumer–resource model
dR/dt = I − β1R/(R01 + R) C1 − β2R/(R02 + R) C2dC1/dt = γ1β1R/(R01 + R) C1 − δ1C1dC2/dt = γ2β2R/(R02 + R) C2 − δ2C2
- R
- the resource (food), renewed at a constant rate I
- C1, C2
- two consumers that only compete by eating the same food
- βi
- the most one consumer can eat per unit time (Holling type II uptake)
- R0i
- the food level at which it eats half of that
- γi
- how much food turns into new consumers
- δi
- death rate per head
Time series
The R* rule
How fast each consumer grows per head at each food level. The food level where it just replaces itself is its R*.
Phase portrait C1–C2
Nine starts and yours (thick), all with the food R(0) set above.
The theory, in short
- Food and eaters. The food is renewed at rate I and eaten; consumers grow in proportion to what they eat (γ) and die at rate δ.
- Holling type II uptake βR/(R0 + R): little food → eating is limited by finding it; plenty → by handling it, up to β.
- R*. The food level where a consumer just replaces itself: R* = δR0/(γβ − δ) (needs γβ > δ).
- Alone, species i settles at R = Ri* with Ci* = Iγi/δi.
- Together, the species with the lower R* wins: it pulls the food down to its own R*, where the other's growth rate is negative (its invasion rate γβR*/(R0 + R*) − δ < 0).
- Competitive exclusion. Two consumers fully dependent on one resource cannot coexist unless R1* = R2* exactly (a fragile tie).
- Not the biggest eater. The winner is the one that survives when food is scarcest, not the one that eats most when it is plenty.
- Fixed points (correcting a sign in the README draft): at equilibrium R*(β1C1/(R01 + R*) + β2C2/(R02 + R*)) = I.