Problem 91 - Calculus for Biology and Medicine

Question

Suppose that a fish population evolves according to a logistic equation and that fish are harvested at a rate proportional to the population size. If \(N(t)\) denotes the population size at time \(t\), then $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)-h N $$ Assume that \(r=2\) and \(K=1000\). (a) Find possible equilibria and use the graphical approach to discuss their stability, when \(h=0.1\). (b) Show that if \(h

Step-by-Step Solution

Answer

The fish population has a stable equilibrium at \( N = 950 \) when \( h = 0.1 \). This equilibrium is stable under \( h < r \), confirmed graphically and with eigenvalue analysis.

Step 1: Understanding the Logistic Equation with Harvesting

The differential equation given is \( \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) - hN \). Here, \( r \) is the intrinsic growth rate, \( K \) is the carrying capacity, and \( h \) is the harvesting rate. The first term represents logistic growth, while the second term represents harvesting at a constant rate proportional to population size.

Step 2: Finding Possible Equilibria for h=0.1

Set the right-hand side of the differential equation to zero to find equilibria: \[ rN \left( 1 - \frac{N}{K} \right) - hN = 0 \] Substituting the values \( r = 2 \), \( K = 1000 \), and \( h = 0.1 \), we have: \[ 2N \left( 1 - \frac{N}{1000} \right) - 0.1N = 0 \] Simplifying, \( N(2 - 0.002N - 0.1) = 0 \), leads to \[ N(1.9 - 0.002N) = 0 \]. Thus, the equilibria are \( N = 0 \) and \( N = 950 \).

Step 3: Analyzing Stability Graphically for h=0.1

Plot the function \( f(N) = 2N(1 - \frac{N}{1000}) - 0.1N \) and note the critical points. Near \( N = 0 \), the slope is positive, indicating it's an unstable equilibrium. Near \( N = 950 \), the function decreases in magnitude as \( N \) moves towards 950. Thus, \( N = 950 \) is a stable equilibrium point.

Step 4: Showing Nontrivial Equilibrium for h < r

For trivial equilibrium, set: \[ 0 = N \times (r - h - \frac{rN}{K}) \]. If \( h < r \), there's a possible positive nontrivial equilibrium. Rearrange \( r - h = \frac{rN}{K} \), yields: \[ N = K \left(1 - \frac{h}{r}\right) \]. For \( h = 0.1, r = 2 \), find \( N = 1000 \left(1 - \frac{0.1}{2}\right) = 950 \).

Step 5: Eigenvalue Analysis for h < r

The derivative of \( f(N) \), \( f'(N) \), evaluates the stability at equilibria. If \( |f'(N)| < 1 \), it's stable. Compute \( f'(N) \) from (3), for \( N = 950 \), substitute values to estimate slope behavior and confirm \( f'(950) < 0 \) to ensure stability.

Step 6: Graphical Stability Analysis Check

Revisit plot: If harvesting rate is below the intrinsic growth, \( N = 950 \) remains stable. \( r-h \) remains positive, indicating stabilizing forces stronger than destabilizing at \( N = 950 \). Parallel comparison for conditions confirms findings.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that describe how a particular quantity changes over time. In the context of population dynamics, these equations can model how a population, like a group of fish, evolves. For the logistic growth equation given in the problem, \[ \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) - hN \]we see two major components:

• The first part, \( rN \left( 1 - \frac{N}{K} \right) \), describes logistic growth. This shows how a population grows quickly at lower densities, slows as the population nears the carrying capacity \( K \), and eventually stabilizes.
• The second component, \( -hN \), represents harvesting. This is the removal or reduction rate that's proportional to the population size. Essentially, the more fish we have, the more are harvested.

This equation helps predict the balance between the natural growth and human interference, providing insights into our impact on ecological systems.

Equilibrium Stability

In differential equations, equilibrium points are where the rate of change is zero, meaning the population doesn't grow or shrink. Finding these points in our exercise means setting the differential equation to zero: \[ rN \left( 1 - \frac{N}{K} \right) - hN = 0 \]Equilibria are significant because they reveal possible steady states of the system where the population remains constant over time. For logistic growth with harvesting, we found two equilibria:

• \( N = 0 \): a trivial equilibrium where no fish exist. This is often unstable as any small disturbances lead to non-zero populations.
• \( N = 950 \): a nontrivial equilibrium reflecting a balance between growth and harvesting.

Stability analysis, particularly through graphical methods or by evaluating the eigenvalues of the system, helps determine whether these points are stable. If we disturb the population slightly, will it return to the equilibrium (stable) or not (unstable)? For example, if \( h < r \), \( N = 950 \) is stable, meaning small deviations correct themselves naturally, maintaining population balance.

Population Dynamics

Population dynamics explores the changes in population size and composition over time, influenced by birth rates, death rates, and factors like harvesting. In our example, the logistic growth model helps to illustrate these dynamics. The intrinsic growth rate \( r \) shows how quickly a population grows without any limiting factors, while \( K \), the carrying capacity, caps the maximum population that the environment can sustain.When we factor in harvesting, represented by \( h \), the population dynamics become more complex. It's crucial to understand:

• How harvesting affects long-term sustainability
• The conditions under which populations either stabilize, explode, or collapse
• The role of equilibria in assessing whether a population is in a precarious spot (like near extinction) or stable

Mathematically analyzing these dynamics helps design strategies for conservation and sustainable resource management, ensuring that populations remain robust despite human interventions.