This problem arises in a model of a plant-herbivore system: (Edelstein-Keshet, 1986). Assume that a population of herbivores of density \(y\) causes changes in the vegetation on which it preys. An internal variable \(x\) reflects some physical or chemical property of the plants which undergo changes in response to herbivory. We refer to this attribute as the plant quality of the vegetation and assume that it may in turn affect the fitness or survivorship of the herbivores. If this happens in a graded, continuous interaction, plant quality may be modeled by a pair of ODEs such as $$ \begin{aligned} &\frac{d x}{d t}=f(x, y) \quad \text { (rate of change of vegetation quality), } \\ &\frac{d y}{d t}=y g(x, y) \quad \text { (herbivore density). } \end{aligned} $$ (a) In one case the function \(f(x, y)\) is assumed to be $$ f(x, y)=x(1-x)[\alpha(1-y)+x] \quad(0 \leq x \leq 1) . $$ Sketch this as a function of \(x\) and reason that the plant quality \(x\) always remains within the interval \((0,1)\) if \(x(0)\) is in this range. Show that plant quality may either decrease or increase depending on (1) initial value of \(x\) and (2) population of herbivores. For a given herbivore population density \(\hat{y}\), what is the "breakeven" point (the level of \(x\) for which \(d x / d t=0) ?\) (b) It is assumed that the herbivore population undergoes logistic growth (see Section 6.1) with a carrying capacity that is directly proportional to current plant quality and reproductive rate \(\beta\). What is the function \(g\) ? (c) With a suitable definition of constants in this problem your equations should have the following nullclines: $$ \begin{array}{ll} x \text { nullclines: } & x=0, \\ & x=1, \\ & x=\alpha(y-1), \\ y \text { nullclines: } & y=0, \\ & y=R x . \end{array} $$ Draw these curves in the \(x y\) plane. (There is more than one possible configuration, depending on the parameter values.) (d) Now find the direction of motion along all nullelines in part (c). Show that under a particular configuration there is a set in the \(x y\) plane that "traps" trajectories. (e) Define \(\gamma=\alpha /(\alpha K-1)\). Interpret the meaning of this parameter. Show that \((\gamma, K \gamma)\) is a steady state of your equations and locate it on your phase plot. Find the other steady state. (f) Using stability analysis, show that for \(\gamma>1,(\gamma, K \gamma)\) is a saddle point whereas for \(\gamma<1\) it is a focus. (g) Now show that as \(\beta\) decreases from large to small values, the steady state \((\gamma<1)\) undergoes the transition from a stable to an unstable focus. (h) Use your results in the preceding parts to comment on the existence of periodic solutions. What would be the biological interpretation of your answer?

Step by step solution

01

- Sketching the function

Consider the function \(f(x, y) = x(1-x)[\alpha(1-y) + x]\). We need to sketch this for \(0 \leq x \leq 1\) and reason why \(x\) remains within this interval. Since \(x(1-x)\) is always positive within \(0 < x < 1\), and assuming \(\alpha(1-y) + x\) is also positive, \(f(x, y)\) will be a quadratic function opening downward.

02

- Analyzing plant quality changes

Plant quality \(x\) changes depend on initial value of \(x\) and herbivore population \(y\). Sketching \(f(x, y)\) shows that changes in \(x\) depend nonlinearly on these parameters. For lower \(\hat{y}\), \(x\) can increase or decrease depending on initial conditions. The breakeven point where \( \frac{dx}{dt} = 0 \) is found by solving \( f(x, y) = 0 \).

03

- Determining the function g(x, y)

Given logistic growth, the density of herbivores \(y\) is proportional to current plant quality \(x\) and reproductive rate \(\beta\). Thus, \(g(x, y) = \beta x (1 - \frac{y}{Kx})\), where \(K\) is the carrying capacity.

04

- Sketching nullclines

The nullclines are given by: for \(x\)-axis, \(x=0\), \(x=1\), and \(x=\alpha(y-1)\); for the \(y\)-axis, \(y=0\) and \(y=R x\). Draw these in the \(xy\) plane to understand the system's equilibrium points.

05

- Finding directions along nullclines

Analyze the direction of motion along the nullclines \(x=\alpha(y-1)\) and \(y=R x\) by determining if \(\frac{dx}{dt} > 0\) or \(\frac{dy}{dt} > 0\) within regions. This helps identify trajectories in the phase plane.

06

- Identifying steady states

Use \(\gamma = \frac{\alpha}{\alpha K - 1}\) to find steady states. The steady state \((\gamma, K \gamma)\) should be located on the phase plot. Find other equilibria by solving for points where both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\).

07

- Stability analysis

Analyze stability for \(\gamma > 1\) and \(\gamma < 1\). For \(\gamma > 1\), the point \((\gamma, K \gamma)\) is a saddle point, while for \(\gamma < 1\), it is a focus. Use Jacobian matrices for precise analysis.

08

- Effect of beta on stability

Show how the decrease in \(\beta\) from large to small values causes the steady state \((\gamma < 1)\) to transition from stable focus to unstable. Use eigenvalues of the Jacobian matrix at the steady state.

09

- Comment on periodic solutions

Based on previous steps, comment on the existence of periodic solutions. The biological interpretation involves the response of herbivore and plant populations to each other, potentially leading to cycles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations

Differential equations are a crucial mathematical tool used to model various phenomena, including plant-herbivore systems. These equations describe how quantities change over time. By setting up differential equations for both plant quality (\(x\)) and herbivore density (\(y\)), we can capture the dynamic interactions in a plant-herbivore system. In our specific example:

• \frac{dx}{dt} describes how the plant quality changes over time, depending on both \(x\) and \(y\).
• \frac{dy}{dt} describes the rate of change of herbivore density, which also relies on \(x\) and \(y\).

These equations can help us understand how initial conditions and external factors influence the long-term behavior of the system. This becomes particularly important when we want to find steady states or study the stability of these states.

Logistic Growth

Logistic growth is a common model used in population dynamics to represent how populations grow in an environment with limited resources. The growth rate is initially exponential when the population is small but slows down as the population approaches the carrying capacity of the environment.
In the context of our plant-herbivore system, the herbivore population (\(y\)) exhibits logistic growth where the carrying capacity is influenced by the current plant quality (\(x\)) and a growth rate parameter (\(\beta\)). The corresponding function for the herbivore growth can be expressed as:
g(x, y) = \beta x \bigg(1 - \frac{y}{Kx}\bigg)
Here:

• \(\beta\) is the reproductive rate of the herbivores.
• \(K\) is the carrying capacity proportional to the plant quality, \(x\).

This equation captures the idea that as plant quality improves, the environment can support more herbivores, while higher herbivore densities can feed back to affect plant quality.

Stability Analysis

Stability analysis is a method used in the study of differential equations to determine whether small deviations from a steady state grow larger over time (unstable) or diminish (stable). In the plant-herbivore model, we perform stability analysis on the steady states identified by solving the differential equations for \frac{dx}{dt} = 0 and \frac{dy}{dt} = 0.
Steps for stability analysis:

• Find the Jacobian matrix of the system of differential equations.
• Evaluate the Jacobian at the steady states.
• Analyze the eigenvalues of the Jacobian to determine the nature of the steady state.

In our example:

• For \frac{\beta}{K-1} > 1, the steady state (\(\frac{\beta}{K-1}\), \(K\frac{\beta}{K-1}\)) is a saddle point, indicating that trajectories move away from this point in some directions.
• For \frac{\beta}{K-1} < 1, the same point is a focus, which could be stable or unstable depending on the eigenvalues.

Understanding these stability properties helps predict how a system responds to perturbations and whether populations in the model tend to return to equilibrium or diverge over time.