Problem 5 - Calculus for Biology and Medicine

Question

Use the graphical approach to classify the following Lotka-Volterra models of interspecific competition according to "coexistence," "founder control," "species 1 excludes species 2," or "species 2 excludes species 1." \(\frac{d N_{1}}{d t}=N_{1}\left(1-\frac{N_{1}}{20}-\frac{N_{2}}{5}\right)\) \(\frac{d N_{2}}{d t}=2 N_{2}\left(1-\frac{N_{2}}{15}-\frac{N_{1}}{3}\right)\)

Step-by-Step Solution

Answer

Species 1 excludes species 2.

Step 1: Identify the Carrying Capacities

Examine the given equations to determine the carrying capacities (the population sizes at which each species experiences no growth when other species are absent). The carrying capacity for species 1 is found when \( \frac{d N_{1}}{d t} = 0 \) and \( N_{2} = 0 \). This yields \( N_{1} = 20 \). For species 2, \( \frac{d N_{2}}{d t} = 0 \) and \( N_{1} = 0 \), leading to \( N_{2} = 15 \).

Step 2: Determine the Zero-Growth Isoclines

For species 1, the zero-growth isocline comes from setting \( \frac{d N_{1}}{d t} = 0 \) which gives \( N_{1} = 20 - 4N_{2} \). For species 2, the zero-growth isocline comes from \( \frac{d N_{2}}{d t} = 0 \), yielding \( N_{2} = 15 - 0.75N_{1} \). These isoclines show where each species' growth rate is zero given the population of the other species.

Step 3: Graph the Isoclines on a Coordinate Plane

Draw the two isoclines on a coordinate plane with \( N_{1} \) on the x-axis and \( N_{2} \) on the y-axis. The line \( N_{1} = 20 - 4N_{2} \) will intercept \( N_{1} \)-axis at 20 and the \( N_{2} \)-axis at 5. The line \( N_{2} = 15 - 0.75N_{1} \) will intercept \( N_{2} \)-axis at 15 and \( N_{1} \)-axis at 20.

Step 4: Analyze the Feasibility of the Equilibrium Points

Graphically examine where the lines intersect. The intersection represents coexistence. Compute the intersection: setting \( N_{1} = 20 - 4N_{2} \) equal to \( N_{2} = 15 - 0.75N_{1} \), solve for \( N_{1} \) and \( N_{2} \). This solution gives the equilibrium point. Calculate it using algebra, an indirect way to determine what happens if only one species dominates.

Step 5: Classify Based on Graphical Interactions

If the isoclines cross and the intersection is feasible (both \( N_{1} \) and \( N_{2} \) are positive), coexistence is possible. If one line completely dominates the other (above the other's carrying capacity), one species excludes the other. In this case, calculate and evaluate the intersection dynamics from calculations in Step 4 to confirm interactions.

Key Concepts

Interspecific Competition

Interspecific competition describes the phenomenon where different species compete with each other for shared resources. This concept is central in ecological studies because it touches upon how species coexist, outcompete, or exclude each other in various environments. In the Lotka-Volterra models, this competition is modeled by equations that represent the negative effect of one species on the growth of another.

• A higher value indicates more intense competition.
• The coefficients beside population sizes in the differential equations represent the strength of competition.

For example, in our given model, the presence of species 2 has a competitive effect on species 1, visible through its coefficient in the equation. This competition can lead to outcomes such as coexistence, one species excluding the other, or founder control, depending on the model's parameters.

Zero-Growth Isoclines

Zero-growth isoclines are important tools used to visually represent and analyze the outcomes of interspecific competition. These lines denote where a species' growth rate becomes zero, meaning populations are stable at that point if no other factors change.

• For species 1, the isocline is where the rate of change is zero, expressed as: \( N_{1} = 20 - 4N_{2} \).
• The same goes for species 2: \( N_{2} = 15 - 0.75N_{1} \).

These isoclines help to visualize competition dynamics. When plotted on a graph, the points at which these isoclines cross can give insights into potential equilibrium states and whether species can coexist or if one will outcompete the other.

Carrying Capacity

Carrying capacity is a fundamental concept for understanding population dynamics in ecology. It signifies the maximum population size of a species that an environment can sustain indefinitely. It's influenced by resources, predation, and interaction with other species.

For the Lotka-Volterra interspecific competition model, carrying capacities are derived from setting the rate of change \( \frac{dN}{dt} = 0 \) while assuming no interspecies competition. In our example:

• Species 1 has a carrying capacity of 20.
• Species 2 has a carrying capacity of 15.

Carrying capacities help establish the baseline for understanding how each species' population would grow in the absence of the other, providing a context for predicting competitive interactions and outcomes.

Equilibrium Points

Equilibrium points occur where both species reach a stable population size, with no net growth or decline. In terms of competition, these points are where the population dynamics balance, resulting in no change in population sizes.

In the context of our Lotka-Volterra model, finding this equilibrium involves solving the system of equations provided by the zero-growth isoclines:

• The intersection of the isoclines determines where equilibrium may occur.
• The point is feasible when it yields positive population sizes for both species.

Graphically, feasible intersections suggest potential coexistence, while infeasible ones hint that one species may dominate or exclude the other. Algebraically solving these intersections can inform whether the dynamics support coexistence or dominance by one species.