Question

This is all one question, sorry it just has a lot of parts.

0.) Explore fit: Use PPLANE to investigate the behavior of the system by plotting the vextor field with the nullclines. Note that in the PPLANE Equation Window, the parameters ( γ, δ). Try different values (between 0 and 1) of the parameters and obeerve the behavor of the system. Keep track of your obeervations or any interesting A.) Nullelines and vector fiekds: Based on your explorations (from number (0)), how do the pararneter values (for parameters α, β, y, δ ) affect the behavior of the system? Are there any qualitative changes as parameter values change? ie are there changes to the number of equilibria or thestability or the behavior of the solution trajectories? Write a few sentences with your observations. Include and refer to plots from PPL.ANE to support your written explanations. B.) Determine the (equations for) nullclines and equilibrium points(s) from the differential C.) How do your answers for (B) rdaLe and compare to your observations des cribed in (A)? D) How do your answers for (B) relate and compare to your results from Part I? What can equations. you conclude about the stability of the equilibrium solution(s)? Write a sentene or two to answer these questions. Interpreting Results E.) Now interpret your obeervations in the context of this situation/problem. Hold the con version rates fixed and describe the elfect of the model if we were to increase or reduce the production rate of SO2. Support your conclusions with either the quantitative work froam Part A or with qualitative techniques (phase plane etc.) F.) Use a vector fidd from PPLANE an aid to explain the results of the model if the original amount of tither compound were zero. Interpret your answer about the sta- bility of the equilibrium solutions and the longterm behavior in the context of this situation/problem. Does the original amount of either compound affect the long term amount of the respective compound?

Here are the equations.

Answer 1

Given, x(t) = gamma/alpha + (x_0 - gamma/alpha)e^-alpha t y(t) = gamma + delta/beta(1 + e^-beta t) + (alpha x_0 - gamma)/beta - alpha(e^-alpha t - e^-beta t) + y_0 e^-beta t Therefore lim_t rightarrow infinity x(t) = lim_t rightarrow infinity {gamma/alpha + (x_0 - gamma/alpha)e^-alpha t} = gamma/alpha lim_t rightarrow infinity y(t) = lim_t rightarrow infinity{(gamma + delta)/beta (1 - e^-beta t) + (alpha x_0 - gamma)/(beta - alpha) (e^-alpha t - e^-beta t) + y_0 e^-beta t = gamma + delta/beta