Let's solve the problem **step by step as requested**. We'll focus on **part (a) and (b)** for now, with explanations after each step. If you want part (c), let me know! --- ## **Step 1: Nondimensionalization (Part a)** Given system: \[ \frac{dN}{dt} = \left( \frac{K_{max} C}{K_n + C} \right) N - \mu N \] \[ \frac{dC}{dt} = -\alpha \left( \frac{K_{max} C}{K_n + C} \right) N - D(C_0 - C) \] **Let's define dimensionless variables:** - Let \( n = \frac{N}{N^*} \) and \( c = \frac{C}{C_0} \), where \( N^* \) and \( C_0 \) are typical scales for \( N \) and \( C \). - Let \( \tau = tD \) (dimensionless time). **Choose scales:** - Let \( N^* \) be arbitrary for now (will cancel out). - \( C_0 \) is the inflow concentration, a natural scale for \( C \). **Change of variables in time derivatives:** \[ \frac{dN}{dt} = \frac{dN}{d\tau} \cdot \frac{d\tau}{dt} = D \frac{dN}{d\tau} \] \[ \frac{dC}{dt} = D \frac{dC}{d\tau} \] **Rewrite equations using new variables:** 1. For \( N \): \[ D \frac{dN}{d\tau} = \left( \frac{K_{max} C}{K_n + C} \right) N - \mu N \] \[ \frac{dN}{d\tau} = \left( \frac{K_{max} C}{D(K_n + C)} - \frac{\mu}{D} \right) N \] Now use \( N = nN^* \), \( C = cC_0 \), \( K_n = k_n C_0 \) (define \( k_n = K_n / C_0 \)), and \( K_{max}/D = k \): \[ \frac{d(nN^*)}{d\tau} = \left( \frac{K_{max} cC_0}{D(k_n C_0 + cC_0)} - \frac{\mu}{D} \right) nN^* \] Divide both sides by \( N^* \): \[ \frac{dn}{d\tau} = \left( \frac{K_{max} c}{D(k_n + c)} - \frac{\mu}{D} \right) n \] Set \( k = \frac{K_{max}}{D} \), \( m = \frac{\mu}{D} \): \[ \frac{dn}{d\tau} = \left( \frac{k c}{k_n + c} - m \right) n \] 2. For \( C \): \[ D \frac{dC}{d\tau} = -\alpha \left( \frac{K_{max} C}{K_n + C} \right) N - D(C_0 - C) \] \[ \frac{dC}{d\tau} = -\frac{\alpha}{D} \left( \frac{K_{max} C}{K_n + C} \right) N - (C_0 - C) \] Substitute dimensionless variables: \[ \frac{d(cC_0)}{d\tau} = -\frac{\alpha}{D} \left( \frac{K_{max} cC_0}{k_n C_0 + cC_0} \right) nN^* - (C_0 - cC_0) \] \[ C_0 \frac{dc}{d\tau} = -\frac{\alpha}{D} K_{max} \frac{c}{k_n + c} n N^* - C_0 (1 - c) \] Divide both sides by \( C_0 \): \[ \frac{dc}{d\tau} = -\frac{\alpha K_{max} N^*}{D C_0} \frac{c}{k_n + c} n - (1 - c) \] Let \( \beta = \frac{\alpha K_{max} N^*}{D C_0} \): \[ \frac{dc}{d\tau} = -\beta \frac{c}{k_n + c} n - (1 - c) \] --- ### **Step 1 Explanation** We have introduced dimensionless variables and parameters to put the system in a simpler, scaled form. This makes the equations easier to analyze and compare across systems with different units. --- ## **Step 2: Find All Steady States (Part b)** Set derivatives to zero for steady states: \[ \frac{dn}{d\tau} = 0, \quad \frac{dc}{d\tau} = 0 \] From first equation: \[ \left( \frac{k c}{k_n + c} - m \right) n = 0 \] So either \( n = 0 \) or \( \frac{k c}{k_n + c} = m \): #### **Case 1: \( n = 0 \)** Plug into second equation: \[ \frac{dc}{d\tau} = - (1 - c) = 0 \implies c = 1 \] So one steady state is \( (n^*, c^*) = (0, 1) \). #### **Case 2: \( n \neq 0 \)** Then \( \frac{k c}{k_n + c} = m \implies k c = m(k_n + c) \implies k c - m c = m k_n \implies c (k - m) = m k_n \implies c = \frac{m k_n}{k - m} \). Now plug into second equation (\( \frac{dc}{d\tau} = 0 \)): \[ -\beta \frac{c}{k_n + c} n - (1 - c) = 0 \implies \beta \frac{c}{k_n + c} n = -(1 - c) \implies n = -\frac{1 - c}{\beta} \frac{k_n + c}{c} \] But since both \( n > 0 \) and \( c > 0 \), this only makes sense if \( c > 1 \). Let's check with our \( c \) expression: \[ c^* = \frac{m k_n}{k - m} \] Plug into above: \[ n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} \] --- ### **Step 2 Explanation** We have found all steady states: - The **trivial steady state**: \( (n^*, c^*) = (0, 1) \) - The **nontrivial steady state**: \[ c^* = \frac{m k_n}{k - m},\qquad n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} \] where \( k > m \) for positivity. --- ## **Final Answers for (a) and (b):** ### **(a) Dimensionless equations:** \[ \boxed{ \begin{aligned} \frac{dn}{d\tau} &= \left( \frac{k c}{k_n + c} - m \right) n \\ \frac{dc}{d\tau} &= -\beta \frac{c}{k_n + c} n - (1 - c) \end{aligned} } \] with \( k = \frac{K_{max}}{D} \), \( k_n = \frac{K_n}{C_0} \), \( m = \frac{\mu}{D} \), \( \beta = \frac{\alpha K_{max} N^*}{D C_0} \). --- ### **(b) Steady states:** \[ \boxed{ \begin{aligned} &\text{Trivial:} \quad (n^*, c^*) = (0, 1) \\ &\text{Nontrivial:} \quad c^* = \frac{m k_n}{k - m}, \quad n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} \end{aligned} } \] --- **Let me know if you want the stability analysis (part c)!**

Let's proceed step-by-step to analyze the steady states and the constraints on parameters for the existence of positive steady states. --- ## **Step 1: Identify constraints for the non-trivial steady state** From the nondimensional steady state analysis, the non-trivial steady state has: \[ c^* = \frac{m k_n}{k - m} \] and \[ n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} \] ### **Explanation:** - For \( c^* \) to be **positive**, numerator and denominator must be positive: \[ c^* > 0 \quad \Rightarrow \quad \frac{m k_n}{k - m} > 0 \] - Since \( k_n, m, k \) are positive parameters (biological meaning), the sign depends on the denominator: \[ k - m > 0 \quad \Rightarrow \quad k > m \] - Also, for \( c^* < 1 \) (since \( c \) is scaled by inflow concentration \( C_0 \)), we need: \[ c^* = \frac{m k_n}{k - m} < 1 \] which implies: \[ \frac{m k_n}{k - m} < 1 \quad \Rightarrow \quad m k_n < k - m \] Rearranged: \[ k - m > m k_n \] --- ## **Step 2: Final constraints on parameters** **To ensure physically meaningful (positive) steady states:** \[ \boxed{ \begin{aligned} &\text{1.} \quad k > m \quad \text{(to have } c^* > 0) \\ &\text{2.} \quad k - m > m k_n \quad \text{(so that } c^* < 1) \\ &\text{3.} \quad n^* > 0 \quad \text{(which requires } 1 - c^* > 0 \text{)} \\ &\quad \Rightarrow \quad c^* < 1 \end{aligned} } \] **Additional conditions:** - Since \( c^* \) is a scaled concentration, physically \( 0 < c^* < 1 \). - The biomass at steady state: \[ n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} > 0 \] which requires \( c^* < 1 \). --- ## **Final Summary:** ### **Constraints on parameters for positive steady states:** \[ \boxed{ \begin{aligned} &\text{Must have} \quad k > m \\ &\text{And} \quad m k_n < k - m \end{aligned} } \] These inequalities ensure the existence of a positive, biologically meaningful steady state. --- **Final note:** - The trivial steady state always exists. - The non-trivial steady state exists **only if** the above constraints are satisfied, which impose relationships among the parameters \( K_{max}, K_n, \mu, D, N^*, C_0 \). If you'd like, I can help analyze the stability of these steady states next!

Certainly! Here's a clear, step-by-step summary of the analysis: --- ### **(a) Dimensionless Form of the System** By defining dimensionless variables and parameters: - \( n = \frac{N}{N^*} \), - \( c = \frac{C}{C_0} \), - \( \tau = t D \), and choosing appropriate scales, the system reduces to: \[ \boxed{ \begin{aligned} \frac{dn}{d\tau} &= \left( \frac{k c}{k_n + c} - m \right) n, \\ \frac{dc}{d\tau} &= -\beta \frac{c}{k_n + c} n - (1 - c), \end{aligned} } \] where: - \( k = \frac{K_{max}}{D} \), - \( k_n = \frac{K_n}{C_0} \), - \( m = \frac{\mu}{D} \), - \( \beta = \frac{\alpha K_{max} N^*}{D C_0} \). --- ### **(b) Steady States and Constraints** **Steady states are found by setting derivatives to zero:** - **Trivial steady state:** \[ \boxed{ (n^*, c^*) = (0, 1) } \] - **Non-trivial steady state:** \[ \boxed{ c^* = \frac{m k_n}{k - m} } \] \[ n^* = \frac{1 - c^*}{\beta} \cdot \frac{k_n + c^*}{c^*} \] --- ### **(c) Constraints on Parameters for Positive Steady States** For the non-trivial steady state to be biologically meaningful (positive biomass and concentration): \[ \boxed{ \begin{aligned} &\text{Require } c^* > 0 \quad \Rightarrow \quad k > m, \\ &\text{And } c^* < 1 \quad \Rightarrow \quad m k_n < k - m, \end{aligned} } \] which ensures: - The denominator \( k - m \) is positive (so \( c^* > 0 \)), - The scaled concentration \( c^* \) is less than 1, - Biomass \( n^* \) is positive. --- **In summary:** - The system's behavior depends on the relationships among \( K_{max} \), \( K_n \), \( \mu \), and \( D \). - The trivial steady state always exists. - The positive, non-trivial steady state exists **only if** \( k > m \) and \( k - m > m k_n \). Let me know if you'd like further analysis on stability or dynamics!