Consider a system of differential equations relating the populations of Vampires

Question

Consider a system of differential equations relating the populations of Vampires(V), Vampire Hunters(H), and Normal Humans(N), Using the previous problems as an example, we can create a simple modal for this system as follows: V'(t) = v_1 V+v_2H+v_3N+v_4VH+v_5+VN+v_6HN+v_7V^2+v_8H^2+v_9N^2 H(t) = h_1V+h_2H+h_3N+h_4VH+h_5VN+h_6HN+h_7V^2+h_8H^2+h_9N^2 N(t) = n_1V+n_1V+n_2H+n_3N+n_4VH+n-5VN+n_6HN+n_7V^2+n_8^2+n_9N^2 In this model, each coefficient v_i, h_i and n_i, 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 v4 were positive, it would means that the interaction of Vampires and Vampire Hunters leads to an increase in the population of Vampires. If we said that n_2 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. The squared terms will all typically be either negative or zero, as they would only be used for logistic growth. Use your understanding of logistic growth, competition models, and predator-prey models to determine whether each of the coefficients above is positive, negative, or 0. Write your answer as +, -, or 0 on the spaces below.

Explanation / Answer

Solution:

Using predator-prey cycle & logistical growth :------ from the way of Vampire Hunters killing Vampires and Vampires killing Normal humans:

If we go with the graph between all three terms

v1 = + v2= 0 v3= - v4= + v5= - v6= - v7= 0 v8= - v9= -

h1 = - h2= + h3= 0 h4= - h5= - h6= 0 h7= - h8= 0 h9= 0

n1 = + n2= + n3= + n4= + n5= + n6= + n7= 0 n8= 0 n9= -

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