Models that are commonly used in fisheries are $$ \frac{d N}{d t}=N g(N) \text {, } $$ where \(g(N)\) is given by $$ \begin{array}{ll} \text { Ricker model: } & g(N)=r e^{-\beta N} . \\ \text { Beverton-Holt: } & g(N)=\frac{r}{\alpha+N} . \end{array} $$ Analyze the behavior of the solutions to these questions. (Assume \(\alpha, \beta\), \(r>0\) ).

The Ricker Model is often used to describe population dynamics in fisheries. It is defined by the growth rate function: \( g(N) = r e^{-\beta N} \). In this model, the population grows exponentially initially but slows down as population size increases. To analyze this model, substitute \( g(N) \) into the differential equation, resulting in \( \frac{dN}{dt} = N r e^{-\beta N} \). This equation suggests that population growth is influenced by the reproductive rate \( r \) and the density-dependent factor \( \beta \). When evaluating equilibrium points where \( \frac{dN}{dt} = 0 \), we find that \( N = 0 \) is the only equilibrium. Stability analysis requires examining the derivative \( \frac{d}{dN} (N r e^{-\beta N}) \), which indicates that the equilibrium at \( N = 0 \) is unstable.

The Beverton-Holt Model is another key model in fisheries. Its growth rate function is: \( g(N) = \frac{r}{\beta + N} \). This model accounts for limited resources by having the growth rate decrease as population size \( N \) increases. Substituting \( g(N) \) into the general differential equation provides \( \frac{dN}{dt} = \frac{r N}{\beta + N} \). Just like the Ricker Model, we analyze this equation to find equilibrium, starting with \( \frac{dN}{dt} = 0 \), giving \( N = 0 \) as an equilibrium point. Conducting a stability analysis involves computing the derivative \( \frac{d}{dN} \frac{r N}{\beta + N} \) which results in it being positive at \( N = 0 \), indicating this equilibrium is also unstable.

Stability Analysis is essential for understanding how populations return to equilibrium after perturbations. For a given model, we analyze the stability of equilibrium points by studying the sign of the derivative of the growth function. If the derivative \( \frac{dg}{dN} \) is positive, small perturbations away from equilibrium will grow, making the equilibrium unstable. Conversely, if the derivative is negative, perturbations will diminish, making the equilibrium stable. This analysis helps predict whether populations will persist, diminish, or oscillate over time. For both the Ricker and Beverton-Holt models discussed, \( N = 0 \) is an unstable equilibrium, suggesting that the population will not return to zero if it is perturbed.

Equilibrium Points represent the state where the population size does not change over time. Mathematically, this is where the rate of change \( \frac{dN}{dt} \) equals zero. To find these points, set the differential equation equal to zero and solve for \( N \). For the Ricker Model, \( \frac{dN}{dt} = N r e^{-\beta N} = 0 \) gives an equilibrium at \( N = 0 \). Similarly, for the Beverton-Holt Model, \( \frac{dN}{dt} = \frac{r N}{\beta + N} = 0 \) also results in \( N = 0 \). Identifying equilibria allows us to understand critical thresholds in population dynamics and inform management strategies.

Differential Equations play a crucial role in modeling biological processes, such as population dynamics in fisheries. These equations describe how a quantity changes over time. General forms like \( \frac{dN}{dt} = N g(N) \) capture the growth rate of populations and how they interact with internal and external pressures. Models like the Ricker and Beverton-Holt help in predicting future population sizes, analyzing stability, and finding equilibrium points. Studying these differential equations improves our understanding of how populations respond to changes, which is vital for conservation and resource management. Integrating knowledge of these equations allows us to develop better strategies for sustaining fish populations.