# (Solved): 2. Consider the competing species model of two species given below; - $$N_{1}$$ and $$N_{2}$$ b ...

2. Consider the competing species model of two species given below; - $$N_{1}$$ and $$N_{2}$$ be the population levels of the two species - $$r_{1}$$ and $$r_{2}$$ be the linear birth rates - $$K_{1}$$ and $$K_{2}$$ be the carrying capacities of the ecosystem - $$b_{12}$$ and $$b_{21}$$ measure the competitive effect of $$N_{2}$$ on $$N_{1}$$, and $$N_{1}$$ on $$N_{2}$$, which are not equal - Population of these two competing species are represented with the following set of differential equitions; $\begin{array}{l} \frac{\mathrm{d} N_{1}}{\mathrm{~d} t}=r_{1} N_{1}\left[1-\frac{N_{1}}{K_{1}}-b_{12} \frac{N_{2}}{K_{1}}\right] \\ \frac{d N_{2}}{d t}=r_{2} N_{2}\left[1-\frac{N_{2}}{K_{2}}-b_{21} \frac{N_{1}}{K_{2}}\right] \end{array}$ (a) Find the equilibrium points for the system (b) Conduct parametric stability analysis for the equilibrium points having at least one of the populations equal to 0 (c) Give all equilibrium points, and their stabilities for the following parameter combinations; - $$r_{1}=0.05, r_{2}=0.08, K_{1}=500, K_{2}=8000, b_{12}=1.5, b_{21}=4, N_{1}(0)=50, N_{2}(0)=1000$$ - $$r_{1}=0.15, r_{2}=0.05, K_{1}=15000, K_{2}=2000, b_{12}=1.5, b_{21}=4, N_{1}(0)=5000, N_{2}(0)=500$$ (d) Construct the given model in Vensim, and simulate it for the conditions given in (c). Give population dynamics for both populations. (e) Verify the equilibrium points and their stability information from part (c) using the Vensim model. (f) Manually draw the phase trajectories of the system according to the first parameter combination. Verify your trajectories using the Vensim model.

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