Herbivore-Algae Predator-Prey Model

Bruce Hannon, U of IL, Urbana, IL 61801, b-hannon@uiuc.edu, June 99.

 

Let us return to the simple predator-prey model to see that even without migration the system can exhibit a wide range of responses, not just a simple population crash. Assume that the prey are algae in a pond on which an herbivore grazes. The data for this problem has been invented. Its input data, parameters and initial conditions would normally be determined by experiment.

The model consists of two main parts, one for the change in the algae population, one for the herbivore. The algae-growth portion of the model we have seen before in various forms. The growth rate is a function of the algal density, ALGAE. This function is monotonic and declining. Algal growth is calculated as the product of the density and the growth rate.

The algae density is reduced through consumption by the herbivore. The consumption per head is a nonlinear function of the algal density: the greater the density, the higher the consumption per head. The consumption rate is simply the product of the number of herbivore and the consumption per head.

The herbivore death rate is determined by their average life span which is a nonlinear function of the consumption per head: the higher the consumption per head, the longer the life span, within limits. Indirectly, the denser the algae, the lower the herbivore death rate.

The herbivore growth rate is a product of the herbivore stock and the fractional herbivore growth rate, FCN HERB GROW. To increase realism of the model, we make FCN HERB GROW a function of the algae density in the previous time period. This is done by producing an additional stock called ALGAE DELAY. In general, it makes sense to represent herbivore behavior in this way. Herbivore gestation time reflects the origin of this lagged behavior.

The first graph shows the wide swings in algal density and herbivore population over time. The second graph, a plot of algal density against the herbivore population, shows the limit cycle resulting from this particular choice of the variables.

 

 Now it is your turn to try changing things. Can you make the herbivore crash and not re–emerge? Try to maximize the herbivore population. Can you do this by adjusting only the variable FCN HERB GROW, without changing the maximum and minimum rates?

HERBIVORE-ALGAE MODEL

ALGAE(t) = ALGAE(t - dt) + (ALGAE_GROWTH - CONSUMPTION) * dt
INIT ALGAE = 210 {Algae per Area}
INFLOWS:
ALGAE_GROWTH = ALGAE*GROWTH_RATE {Algae per Area per Time Period}
OUTFLOWS:
CONSUMPTION = HERBIVORE*CONSUMP_PER_HD {Algae per Area per Time Period}

HERBIVORE(t) = HERBIVORE(t - dt) + (HERB_GROWTH_RATE - DEATH_RATE) * dt

INIT HERBIVORE = 45 {Individuals}
INFLOWS:
HERB_GROWTH_RATE = HERBIVORE*FCN_HERB_GROW {Individuals per Time Period}
OUTFLOWS:
DEATH_RATE = HERBIVORE/LIFESPAN {Individuals per Time Period}

ALGAE_DELAY = DELAY(ALGAE,2) {Individuals}
CONSUMP_PER_HD = GRAPH(ALGAE)
(0.00, 0.00), (100, 0.25), (200, 0.6), (300, 0.83), (400, 1.06), (500, 1.24), (600, 1.41), (700, 1.61), (800, 1.77), (900, 1.89), (1000, 1.98)
FCN_HERB_GROW = GRAPH(ALGAE_DELAY)
(0.00, 0.00), (100, 0.0035), (200, 0.0075), (300, 0.019), (400, 0.065), (500, 0.13), (600, 0.163), (700, 0.181), (800, 0.19), (900, 0.195), (1000, 0.198)
GROWTH_RATE = GRAPH(ALGAE)
(0.00, 0.21), (100, 0.168), (200, 0.112), (300, 0.0902), (400, 0.0781), (500, 0.066), (600, 0.0572), (700, 0.0462), (800, 0.0363), (900, 0.0198), (1000, 0.00)
LIFESPAN = GRAPH(CONSUMP_PER_HD)
(0.00, 0.00), (0.2, 2.16), (0.4, 4.32), (0.6, 6.96), (0.8, 9.48), (1.00, 12.1), (1.20, 14.9), (1.40, 17.3), (1.60, 20.2), (1.80, 22.6), (2.00, 23.8)