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Chapter 8: Problem 10

(Budworms vs. the forest) Ludwig et al. (1978) proposed a model for the effects of spruce budworm on the balsam fir forest. In Section 3.7, we considered the dynamics of the budworm population; now we turn to the dynamics of the forest. The condition of the forest is assumed to be characterized by \(S(t)\), the average size of the trees, and \(E(t)\), the "energy reserve" (a generalized measure of the forest's health). In the presence of a constant budworm population \(B\), the forest dynamics are given by $$ \dot{S}=r_{5} S\left(1-\frac{S}{K_{5}} \frac{K_{E}}{E}\right), \quad \bar{E}=r_{E} E\left(1-\frac{E}{K_{\bar{E}}}\right)-P \frac{B}{S} $$ where \(r_{5}, r_{E}, K_{5}, K_{E}, P>0\) are parameters. a) Interpret the terms in the model biologically. b) Nondimensionalize the system. c) Sketch the nullclines. Show that there are two fixed points if \(B\) is small, and none if \(B\) is large. What type of bifurcation occurs at the critical value of \(B\) ? d) Sketch the phase portrait for both large and small values of \(B\).

Short Answer

Expert verified
The given model describes the effects of spruce budworms on the balsam fir forest, with \(S(t)\) representing the average tree size and \(E(t)\) representing the energy reserve. The model is nondimensionalized using new variables \(s\) and \(e\). The nullclines are found, and it is shown that there are two fixed points for small values of \(B\) and none for large values of \(B\). This indicates a Saddle-Node bifurcation occurs at the critical value of \(B\). The phase portraits are sketched for both large and small values of \(B\), showing a stable forest for small values and a collapsing forest for large values.

Step by step solution

01

Interpreting the terms in the model biologically

1. \(S(t)\): Represents the average tree size at time t. 2. \(E(t)\): Represents the energy reserve (measure of the forest's health) at time t. 3. \(B\): Represents the spruce budworm population. 4. \(r_5\): Represents the intrinsic growth rate of the average tree size. 5. \(r_E\): Represents the intrinsic growth rate of the energy reserve. 6. \(K_5\): Carrying capacity for tree size. 7. \(K_E\): Carrying capacity for the energy reserve. 8. \(K_{\bar{E}}\): Energy reserve threshold beyond which it cannot grow. 9. \(P\): Represents the rate at which the budworm population affects the forest's energy reserve.
02

Nondimensionalize the system

To nondimensionalize the system, we introduce new variables: Let \(s = \frac{S}{K_5}\) and \(e = \frac{E}{K_{\bar{E}}}\), where \(s\) and \(e\) are dimensionless. Now rewrite the system in terms of \(s\) and \(e\): $$ \begin{aligned} \dot{s} &= \frac{1}{K_{5}} \dot{S} = r_5 \frac{S}{K_5} \left(1-\frac{S}{K_{5}}\frac{K_{E}}{E}\right) = r_5 s \left(1 - s\frac{K_{E}}{eK_{\bar{E}}}\right), \\ \dot{e} &= \frac{1}{K_{\bar{E}}}\dot{E} = \frac{r_{E} E}{K_{\bar{E}}}\left(1-\frac{E}{K_{\bar{E}}}\right) - \frac{P}{K_{\bar{E}}} \frac{B}{S} = r_E e (1 - e) - \frac{P}{K_{\bar{E}}}\frac{B}{K_5 s}. \end{aligned} $$ Let \(\tilde{B} = \frac{BP}{K_{\bar{E}}K_5}\) and \(\beta = \frac{K_E}{K_{\bar{E}}}\). Then, the nondimensional system is: $$ \begin{aligned} \dot{s} &= r_5 s \left(1-s\frac{\beta}{e}\right), \\ \dot{e} &= r_E e (1-e) - \tilde{B}\frac{1}{s}. \end{aligned} $$
03

Sketch nullclines and find fixed points

To find the nullclines, set the derivatives equal to zero: For \(s\)-nullcline: $$ 0 = r_5 s \left(1 - s\frac{\beta}{e}\right) $$ For \(e\)-nullcline: $$ 0 = r_E e (1 - e) - \tilde{B}\frac{1}{s} $$ To find the fixed points, solve the nullcline equations simultaneously: a) Two fixed points if \(B\) is small: Let \(0 < \tilde{B} < r_{E}\), then there exist positive values of \(s\) and \(e\) satisfying the above system of equations, which gives two fixed points. b) No fixed points if \(B\) is large: If \(\tilde{B} > r_{E}\), there is no positive \(s\) and \(e\) satisfying the above system of equations.
04

Type of bifurcation

As the number of fixed points in the system changes with parameter \(\tilde{B}\), this system undergoes a Saddle-Node bifurcation at the critical value of \(\tilde{B}\).
05

Sketch phase portraits

To sketch the phase portrait, we analyze the behavior of the system for large and small values of \(\tilde{B}\): a) For small values of \(\tilde{B}\): We have two fixed points. Depending on the parameters, one fixed point can be a stable node or spiral (where the forest thrives) and the other fixed point can be an unstable saddle (where the forest collapses). b) For large values of \(\tilde{B}\): No fixed points exist, and the forest collapses as \(\dot{s}\) and \(\dot{e}\) are negative for all positive values of \(s\) and \(e\).

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Key Concepts

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

Bifurcation Theory
In the context of nonlinear dynamics, bifurcation theory studies how a small change in the system's parameters can cause a sudden and dramatic shift in its behavior. In our budworm and forest model, the parameter of interest is the budworm population, denoted as \( B \).
When budworm population \( B \) is small, our system has two fixed points, which represent different stable states of the forest. These fixed points are found by analyzing where the derivatives \( \dot{s} \) and \( \dot{e} \) of the model equations are zero. As \( B \) increases to a critical value, these two fixed points merge and then disappear.
This behavior is characteristic of a Saddle-Node bifurcation, where an attractor and a repeller annihilate each other. As a result, beyond this critical value, the system shifts to a new state, potentially leading to the collapse of the forest. This phenomenon neatly illustrates how seemingly minor changes in environmental stressors, like budworm population, may critically impact forest health.
Population Dynamics
Population dynamics explore how and why populations change over time. In the forest-budworm model, the main focus is on understanding changes in the forest's state due to varying budworm populations.
The spruce budworm are pests that adversely affect the balsam fir forest by feeding on it. Their population, denoted as \( B \), plays a crucial role in determining the competitive dynamics of the forest ecosystem.
When \( B \) is small, the forest can support regeneration and maintain its health through its intrinsic growth rates \( r_5 \) and \( r_E \). However, with high values of \( B \), the negative impact of the budworms on tree size and energy reserve hinders growth, leading potentially to a critical environmental shift. Such dynamics emphasize the delicate balance in ecosystems, where certain species populations can lead to large-scale environmental changes.
Mathematical Modeling
Mathematical modeling provides a framework to represent complex systems using equations to predict their behavior under various conditions. In this case, we model the forest dynamics affected by the spruce budworm population.
The equations we use describe how the average tree size \( S(t) \) and energy reserve \( E(t) \) change over time. These equations incorporate several ecological parameters, such as growth rates \( r_5 \) and \( r_E \), and carrying capacities \( K_5 \) and \( K_{\bar{E}} \).
By nondimensionalizing the system, we make the equations simpler and easier to analyze, reducing the complexity of the problem to finding how the ratios of \( S \) and \( E \) to their respective carrying capacities evolve.
  • The model helps us visualize different scenarios through phase portraits, providing insights into potential stable states (fixed points) of the forest.
  • Through these models, we can predict outcomes like forest collapse under large budworm populations, offering valuable insight for conservation strategies.
Ultimately, mathematical modeling grants us powerful tools to both grasp and tackle real-world ecological challenges.

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Most popular questions from this chapter

For each of the following systems, a Hopf bifurcation occurs at the origin when \(\mu=0\). Using a computer, plot the phase portrait and determine whether the bifurcation is subcritical or supercritical. $$ \dot{x}=y+\mu x, \quad y=-x+\mu y-x^{2} y $$

Reconsider the system (8.6.1): $$ \dot{\theta}_{1}=\omega_{1}+K_{1} \sin \left(\theta_{2}-\theta_{1}\right), \quad \dot{\theta}_{2}=\omega_{2}+K_{2} \sin \left(\theta_{1}-\theta_{2}\right) $$ a) Show that the system has no fixed points, given that \(\omega_{1}, \omega_{2}>0\) and \(K_{1}\), \(K_{2}>0\) b) Find a conserved quantity for the system. (Hint: Solve for \(\sin \left(\theta_{2}-\theta_{1}\right)\) in two ways. The existence of a conserved quantity shows that this system is a nongeneric flow on the torus; normally there would not be any conserved quantities_ ) c) Suppose that \(K_{1}=K_{2}\). Show that the system can be nondimensionalized to $$ d \theta_{1} / d \tau=1+a \sin \left(\theta_{2}-\theta_{1}\right), \quad d \theta_{2} / d \tau=\omega+a \sin \left(\theta_{1}-\theta_{2}\right) $$ d) Find the winding number \(\lim _{f \rightarrow \infty} \theta_{1}(\tau) / \theta_{2}(\tau)\) analytically. (Hint: Evaluate the Iong-time averages \(\left\langle d\left(\theta_{1}+\theta_{2}\right) / d \tau\right\rangle\) and \(\left\langle d\left(\theta_{1}-\theta_{2}\right) / d \tau\right\rangle\), where the brackets are defined by \((f)=\lim _{T \rightarrow-}+\int_{0}^{T} f(\tau) d \tau\), For another approach, see Guckenheimer and Holmes \((1983,0.299)\).)

(Predator-prey model) Odell (1980) considered the system $$ \dot{x}=x[x(1-x)-y], \quad \hat{y}=y(x-a) $$ where \(x \geq 0\) is the dimensionless population of the prey, \(y \geq 0\) is the dimension- less population of the predator, and \(a \geq 0\) is a control parameter. a) Sketch the nullelines in the first quadrant \(x, y \geq 0\). b) Show that the fixed points are \((0,0),(1,0)\), and \(\left(a, a-a^{2}\right)\), and classify them. c) Sketch the phase portrait for \(a>1\), and show that the predators go extinct. d) Show that a Hopf bifurcation occurs at \(a_{e}=\pm .\) Is it subcritical or supercritical? e) Estimate the frequency of limit cycle oscillations for \(a\) near the bifurcation. f) Sketch all the topologically different phase portraits for \(0

(Irrational flow yields dense orbits) Consider the flow on the torus given by \(\dot{\theta}_{1}=\omega_{1}, \dot{\theta}_{2}=\omega_{2}\), where \(\omega_{1} / \omega_{2}\) is irrational. Show each trajectory is dense; i.e., given any point \(p\) on the torus, any initial condition \(q\), and any \(\varepsilon>0\), there is some \(t<\infty\) such that the trajectory starting at \(q\) passes within a distance \(\varepsilon\) of \(p\).

(Laser model) In Fxercise \(3.3 .1\) we introduced the laser model $$ \begin{aligned} &\dot{n}=G n N-k n \\ &\dot{N}=-G n N-f N+p \end{aligned} $$ where \(N(t)\) is the number of excited atoms and \(n(t)\) is the number of photons in the laser field. The parameter \(G\) is the gain coefficient for stimulated emission, \(k\) is the decay rate due to loss of photons by mirror transmission, scattering, etc., \(f\) is the decay rate for spontaneous emission, and \(p\) is the pump strength. All parameters are positive, except \(p\), which can have either sign. For more information, see Milonni and Eberly (1988). a) Nondimensionalize the system. b) Find and classify all the fixed points. c) Sketch all the qualitatively different phase portraits that occur as the dimensionless parameters are varied. d) Plot the stability diagram for the system. What types of bifurcation occur?
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