Let's solve this step-by-step as requested. We'll do **parts (a) and (b)** here, with explanations after each step. --- ### Step 1: (a) Why does it make sense that \(x \ge 0\) and \(y \ge 0\) in this system? **Explanation:** - The variables \(x\) and \(y\) represent the populations of two species. - Populations cannot be negative in real-world scenarios—there can't be a negative number of organisms. - Therefore, it makes sense to restrict \(x \ge 0\) and \(y \ge 0\). --- ### Step 2: (b) Find the \(x\)-nullclines. Recall: The \(x\)-nullcline is where \(\frac{dx}{dt} = 0\). Given: \[ \frac{dx}{dt} = xy - 4x + x^3 \] Set to zero: \[ xy - 4x + x^3 = 0 \] \[ x(y - 4 + x^2) = 0 \] So, this is zero when: 1. \(x = 0\) 2. \(y - 4 + x^2 = 0 \implies y = 4 - x^2\) **Explanation:** - The \(x\)-nullclines are the set of points where the population \(x\) does not change. - These occur along the lines \(x = 0\) and \(y = 4 - x^2\). --- **Final Answer for Steps 1 & 2:** **(a)** It makes sense that \(x \ge 0\) and \(y \ge 0\) because \(x\) and \(y\) represent populations, which cannot be negative. **(b)** The \(x\)-nullclines are: - \(x = 0\) - \(y = 4 - x^2\)

Let's proceed step-by-step as required. --- ### **Step 1: Why does it make sense that \(x \ge 0\) and \(y \ge 0\)?** **Explanation:** - The variables \(x\) and \(y\) represent populations of two species in an ecological model. - Populations cannot be negative in reality; negative values would not make biological sense. - Therefore, the model is valid in the region where \(x \ge 0\) and \(y \ge 0\), ensuring the populations stay within physically meaningful bounds. --- ### **Step 2: Find the \(x\)-nullclines** **Given:** \[ \frac{dx}{dt} = xy - 4x + x^3 \] Set \(\frac{dx}{dt} = 0\): \[ xy - 4x + x^3 = 0 \] Factor out \(x\): \[ x(y - 4 + x^2) = 0 \] **Solutions:** - \(x = 0\), or - \(y - 4 + x^2 = 0 \Rightarrow y = 4 - x^2\) **Explanation:** - The \(x\)-nullclines are the sets where the rate of change of \(x\) is zero. - These occur along the line \(x=0\) (the \(y\)-axis), representing scenarios where the population \(x\) is zero. - They also occur along the parabola \(y=4 - x^2\), where the growth rate of \(x\) is zero due to balancing effects. --- ### **Final answers:** - **(a)** It makes sense that \(x \ge 0\) and \(y \ge 0\) because these variables model populations, which are inherently non-negative. - **(b)** The \(x\)-nullclines are given by: \[ \boxed{ x=0 \quad \text{and} \quad y=4 - x^2 } \] --- Let me know if you'd like to proceed with the \(y\)-nullclines or further analysis!