Question

In the 1920 's, the Italian mathematician Umberto Volterra proposed the following mathematical model of a predator-prey situation to explain why, during the first World War, a larger percentage of the catch of Italian fishermen consisted of sharks and other fish eating fish than was true both before and after the war. Let \(x(t)\) denote the population of the prey, and let \(y(t)\) denote the population of the predators. In the absence of the predators, the prey population would have a birth rate greater than its death rate, and consequently would grow according to the exponential model of population growth, i.e. the growth rate of the population would be proportional to the population itself. The presence of the predator population has the effect of reducing the growth rate, and this reduction depends on the number of encounters between individuals of the two species. Since it is reasonable to assume that the number of such encounters is proportional to the number of individuals of each population, the reduction in the growth rate is also proportional to the product of the two populations, i.e., there are constants \(a\) and \(b\) such that $$ x^{\prime}=a x-b x y $$ Since the predator population depends on the prey population for its food supply it is natural to assume that in the absence of the prey population, the predator population would actually decrease, i.e. the growth rate would be negative. Furthermore the (negative) growth rate is proportional to the population. The presence of the prey population would provide a source of food, so it would increase the growth rate of the predator species. By the same reasoning used for the prey species, this increase would be proportional to the product of the two populations. Thus, there are constants \(c\) and \(d\) such that $$ y^{\prime}=-c y+d x y . $$ a) A typical example would be with the constants given by \(a=0.4, b=0.01, c=0.3\), and \(d=0.005\). Start with initial conditions \(x_{1}(0)=50\) and \(x_{2}(0)=30\), and compute the solution to (8.22) and (8.23) over the interval \([0,100]\). Prepare both a time plot and a phase plane plot. After Volterra had obtained his model of the predator-prey populations, he improved it to include the effect of "fishing," or more generally of a removal of individuals of the two populations which does not discriminate between the two species. The effect would be a reduction in the growth rate for each of the populations by an amount which is proportional to the individual populations. Furthermore, if the removal is truly indiscriminate, the proportionality constant will be the same in each case. Thus, the model in equations \((8.22)\) and (8.23) must be changed to $$ \begin{aligned} &x^{\prime}=a x-b x y-e x \\ &y^{\prime}=-c y+d x y-e y \end{aligned} $$ where \(e\) is another constant. b) To see the effect of indiscriminate reduction, compute the solutions to the system in \((8.24)\) when \(e=0\), \(0.01,0.02,0.03\), and \(0.04\), and the other constants are the same as they were in part a). Plot the five solutions on the same phase plane, and label them properly. c) Can you use the plot you constructed in part b) to explain why the fishermen caught more sharks during World War I? You can assume that because of the war they did less fishing.

Short answer

Reduced fishing during WWI (smaller \(e\)) led to stable predator populations like sharks.

Step-by-step solution

Understand the Mathematical Model

The predator-prey model equations are given by two differential equations. In the absence of predators, the prey (\(x(t)\)) grows exponentially, reduced by predator interactions, leading to \(x'(t) = ax - bxy\). Conversely, predators depend on prey and their growth rate is negative without prey, modified by presence of prey as \(y'(t) = -cy + dxy\).

Analyze Initial Conditions and Parameters

Given are initial conditions \(x(0) = 50\) and \(y(0) = 30\), with parameters \(a = 0.4\), \(b = 0.01\), \(c = 0.3\), and \(d = 0.005\). These will help simulate the predator-prey interactions over a certain period.

Solve the Equations Over the Interval [0, 100]

Use numerical methods (e.g., the Runge-Kutta method) to solve the differential equations from time 0 to 100. Calculate populations at different time intervals.

Plot Time and Phase Portraits for Part a

Create a time plot showing both predator and prey populations over time. Also, create a phase plane plot with \(x(t)\) on the x-axis and \(y(t)\) on the y-axis to visualize the interaction between the two populations.

Modify the Equations for Indiscriminate Reduction

Introduce the constant \(e\) for indiscriminate reduction, altering the original system to: \(x'(t) = ax - bxy - ex\) and \(y'(t) = -cy + dxy - ey\).

Compute Solutions for Various e-Values

Solve the modified equations for \(e = 0, 0.01, 0.02, 0.03, 0.04\) individually using initial conditions and parameters from Part a (excluding \(e\)).

Plot Phase Plane for Different e-Values

Create a phase plane plot for each \(e\) value obtained in Step 6. Compare the behavior of predator-prey interaction for changing values of \(e\).

Interpretation and Explanation for More Sharks in Part c

Increased shark catches during WWI can be explained by less fishing due to the war, corresponding to a decrease in indiscriminate reduction (i.e., smaller \(e\) value), enhancing predator population stability during that time.

Key concepts

Volterra equations

The Volterra equations, named after the mathematician Umberto Volterra, describe the interactions between predator and prey species in ecological systems. These equations form a pair of linked first-order differential equations and serve as a mathematical model to predict the population changes over time.
Volterra's model is expressed in the following way for prey and predator populations:

• The rate of change of the prey population, represented as \( x'(t) = ax - bxy \), where \( x(t) \) is the prey population size, \( a \) is the prey birth rate, and \( b \) is the rate of predation reliant on prey and predator interaction.
• The rate of change of the predator population is represented as \( y'(t) = -cy + dxy \), where \( y(t) \) is the predator population, \( c \) is the predator death rate, and \( d \) is the increase in predator population due to consuming prey.

These equations highlight the interdependence between the two species, showcasing how changes in one population directly impact the other.

differential equations

Differential equations play a critical role in modeling dynamic systems, especially in the context of the predator-prey model. Such equations are mathematical expressions that represent rates of change and their solutions illustrate how one quantity evolves relative to another.
In the predator-prey scenario, the differential equations \( x'(t) = ax - bxy \) and \( y'(t) = -cy + dxy \) capture the growth and decline of prey and predator populations over time.

• The equation for prey takes into account both natural population growth (proportional to \( a \)) and reduction due to predation at a rate \( bxy \).
• For predators, the equation considers decline through natural death rates (\( -cy \)) mitigated by feeding on prey, as indicated by the term \( dxy \).

Solving these equations typically involves computational methods such as the Runge-Kutta technique, to simulate how populations would evolve over a specified time frame.

phase plane analysis

Phase plane analysis provides a visual representation of the dynamic behavior of systems described by differential equations. In the context of predator-prey models, this method maps the interaction of prey and predator populations on a graph where each axis corresponds to one of the species.
Typically, the phase plane plot depicts the prey population \( x(t) \) on the horizontal axis and the predator population \( y(t) \) on the vertical axis.

• Trajectories in the phase plane indicate the path that the populations follow over time, capturing cycles like predator boom and bust phases linked to prey availability.
• By observing these trajectories, one can analyze stability, cyclic patterns, and other behaviors.

Phase plane analysis is instrumental in understanding predator-prey interactions without direct time dependence, allowing for insights into how populations dynamically adjust around equilibrium points.

population dynamics

Population dynamics study the changes in population sizes and compositions in ecosystems. It's driven by biological and environmental factors like birth rates, death rates, competition, disease, and predation, as seen in the predator-prey model.
In this context, understanding the balance between predator and prey populations provides insight into ecological health and resource availability.

• The prey's potential for exponential growth can lead to population booms, depending on birth rates exceeding death rates in a predator-free environment.
• Conversely, predators regulate prey numbers, balancing the ecosystem by keeping prey growth in check through natural predation.

In the real world, factors such as human activities, exemplified by indiscriminate fishing, can disrupt these dynamics, leading to shifts in population stability both during and post-intervening events, influencing ecological balance.