Spatial
Predator-Prey Models
The Basic Model of Predator-Prey Interactions Bruce Hannon, U of IL, Urbana, IL 61801, b-hannon@uiuc.edu, June 99. In the ecological models of the previous chapters much attention is given to the dynamics of a single population. In contrast, in this chapter we concentrate on the dynamics of interspecies interactions. The two species model consists of a predator population and a prey population. Both are present within a given area with a specified carrying capacity. The birth rates for predators and prey are given exogenously, yet the number of births depends on predator-prey interaction and the carrying capacity of their ecosystem. Similarly, the deaths of predators and prey depend on their interaction - the prey consumed by predators. Run the model for different initial population sizes, birth and death rates, and carrying capacities. Can you save the populations from crashing?
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Full Spatial Predator-Prey Model The model in the previous section had all predators and prey in the same place. There, the populations of predators and prey always crashed. How would our result differ if we broke up the same area into nine subcells, each with 1/9th of the original carrying capacity. All other model features held the same, but now, we have predators and prey move among the cells, "chasing or fleeing from each other". The prey migrate routinely regardless of their population in the starting or the receiving cells, and the predators migrate to a new cell when they begin to starve in their current cell. Once the migration quantity is established, a random process determines its distribution to adjacent cells. This simple idea enables the prey to "escape" to a neighboring cell where the predator population may be at a relatively low level. But the cellular model is more realistic than the single cell model; it adds geographic reality to the problem and it allows differentiation of predator and prey behavior. The STELLA II model for the first cell of the 9 cell predator-prey model is shown in the following diagram. This first cell is in the upper left hand corner of the square 9 cell model. Consequently, predators and prey can move to the right (RT) and down (DN). The fraction of predators and prey that move is a random number between zero and one. The migration rate of prey, MIG RATE PREY, is specified exogenously. The number of prey consumed in cell 1 cannot exceed the number of prey in that cell and is at least zero and at most the consumption rate times the number of predators in that cell. Thus, CONSUME__1 = MIN(PREY_1,MAX(0,CONSUME_RATE*PRED_1)). The consumption rate, in turn, is also given exogenously. The migration rate of predators, MIG RATE PRED, depends on the availability of prey in the cell. For simplicity, we assume that the number of predators that migrate is the product of the number of starvation deaths and the migration rate of predators. A listing of the STELLA II equations for the first cell of the 9 cell model is given at the end of this chapter. In order to build the 9 cell model the diagram is just duplicated and slightly modified. The modification necessary includes, besides a change of variable names, rules that enable movement in the proper directions. The nine parts of the spatial predator-prey model can be combined to visually represent the predator and prey populations of each cell at a point in time. Develop "Ghosts" in STELLA II and animate them so that when you run the model you can see the population sizes change for each cell. Also, calculate the total populations of predators and prey and plot them over time. Run this model and you will find that it will crash only very rarely. Try changing the parameters again and you will find that it is difficult to make it crash! Trim the number of cells to four, spread the initial population evenly and see what happens. Try the four cells in a row and then in a square. Do you find a difference in survival times? Does this step give you any ideas about the shape and size of residue habitat which can maximize survival probability. Now try five in a row vs. one cell with four cells arranged on each its sides. Run your models with different DT. How do the results differ, and why? If the model were expanded to 33x33 cells, would we expect to see patches with only prey, others with only predators? By noting the largest range of the predators in a natural area, one could test the assemblages needed to meet prescribed probabilities of survival. Presumably, this largest range would enable the remainder of the living components of the system to survive easily. What are the basic variables here? Cellular configuration, initial predator-prey population and the distribution of the variation of the carrying capacity. Try to eliminate the latter two in order to see what real difference the configuration of the cell can make to the survival of the predator-prey process.
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