Consider the single species population model defined by
$$\frac{dR}{dt} = \frac{gR}{R+R_m} - dR,$$
for $t > 0$, where $g,R_m$, and $d$ are all positive parameters and $R(0) =R_0$.
(a) Describe the biological meaning of each term in the equation.
(b) Determine the steady-states of the system and discuss any constraints on the model parameters for the model to admit biologically meaningful solutions.
(c) Determine the steady-state stability and discuss any variation in this with respect to the model parameter values.
=> a) $gR$ represents the exponential growth of population, $dR$ represents the exponential decay of population, $g$ is the growth rate, and $d$ is the decay rate. What do $R$ and $R_m$ represent? How can I define the term $$\frac{gR}{R+R_m}?$$ What is $R+R_m$? Does it affect the $gR$ for the grow?
b) In single steady-state system, $$\frac{dR}{dt} =0.$$ $$\frac{gR}{R+R_m} - dR = 0$$ $$ gR -dR(R+R_m) =0 $$ $$ R(g -d(R+R_m)) =0$$ either $R=0$ or $g -d(R+R_m)= 0$. $$g- dR_m =0 \ \ \ (R=0)$$ $$R_m = \frac{g}{d}$$ $$R^{\ast} = \frac{g}{d} \ \ \ (R_m = R^{\ast})$$ So, we have $R^{ast}_1 = 0$, $R^{\ast}_2 = \frac{g}{d}$ Are these correct? I am not sure constraint on the model parameters to admit biologically meaning solutions?
c) to determine steady-state stability let $$f(R) =\frac{gR}{R+R_m} - dR $$ $$\frac{df}{dR} = g ln(R+R_m) -d . $$ My differentiation may be wrong and don't know the term $R_m$ while differentiating with respect to $R$. I really don't know after that. and I know my answer is still incomplete.
$R_m$ is a constant. Your computation of the steady states doesn't look correct. $R=0$ is one steady state, and solving $g - d(R + R_m) = 0$ for $R$ will give you the other. When computing $\frac{df}{dR}$, you should use the quotient rule. Again, $R_m$ is a constant so its derivative is zero.