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21. Consider a system of differential equations relating the populations of Vampires (V), Vampire Hunters (H), and Normal Humans (N). Building on our understanding of Predator-Prey models, Logistic Growth models, and Competition Models, we can create a simple model for this system as follows:
V(t) = Vo + vV + vH + vN + vVH + vsVN + v6HN + vV^2
H(t) = ho + hV + hH + hN + hVH + hsVN + hHN + hH^2
N(t) = no + nV + nH + nN + nVH + nVN + nHN + nN
In this model, each coefficient v, h, and n will represent either a positive number, a negative number, or a zero. If it is positive, it means that the population or interaction will have a positive effect. If it is negative, it means that the particular population or interaction will have a negative effect. If it is zero, it means that the particular population or interaction will have no effect at all on the population.
As an example, if we said that v were positive, it would mean that the interaction of Vampires and Vampire Hunters leads to an increase in the population of Vampires. If we said that n were negative, it would mean that if there were no Normal Humans and no Vampires, the population of Normal people would be decreasing at a rate proportional to the number of Vampire Hunters. If we said that Vo were positive, it would mean that if all three populations were zero, we would still naturally have a linear increase in the number of vampires.
The squared terms will all typically be either negative or zero, as they would only be used for logistic growth.
a) Use your understanding of logistic growth, competition models, and predator-prey models to determine whether each of the coefficients above is positive, negative, or zero. (As vampires are likely fictional, answers may vary.)
b) Suppose that instead of modeling Vampires, V represents the population of a Virus which attacks Red Blood Cells, H represents the population of White Blood Cells, and N represents the population of Red Blood Cells. Explain in words how your model might be different.
