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The standard Lotka-Volterra equations for predator-prey systems are dN N =rN (1 - K - G₁NP₁ - ₂NP₂ dP₁ =&C₁NP₁-Z₁P₁ dt dP₂ == & C₂NP₂-Z₂P₂ dt where r is population growth rate for the prey, K is the prey's carrying capacity, C₁ and C₂ are the consumption rate for two different predator populations with number densities P₁ and P₂ that both consume the prey population, & is the efficiency of converting prey biomass into predator biomass, Z₁ and Z₂ are the intrinsic mortality rate for the two predator species respectively, and N is the number density of the prey species. a. Write down the interaction matrix, A, for this system. What is the network diagram for this set of equations? b. Find the fixed points of the system. How many are nontrivial (i.e., not all species have zero abundance)? c. What is the Jacobian for the system evaluated at a fixed point where the P₂ predator is not zero and thus not extinct? d. Is the Trace always positive or always negative or does it depend on the parameter values? Based on your answer to problem 2 for how the Trace relates to the stability of the system, interpret how each parameter contributes to the sign of the trace and argue how this makes biological sense in terms of whether the system will return to this fixed point. e. If the P₂ species is not a predator but instead is a mutualist species with the prey so that it benefits the prey and the prey benefits it, how would you modify the above equations to account for this? dt