Evolutionary Dynamics and the Ludwig-Jones-HollingDifferential Equation for Spruce Budworm Populations

Item Type text; Electronic Thesis

Authors Tatem, Alexia Rae

Publisher The University of Arizona.

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Abstract

This paper studies the Ludwig-Jones-Holling differential equations model in the presence of evolution. To our knowledge, this famous model has not been studied in the context of evolutionary adaption. We apply evolutionary game theory to obtain a system of differential equations for a spruce budworm population x and a mean phenotypic trait u. There are several parameters on which the system depends and we make various assumptions regarding the parameters as they depend on the mean phenotypic train u. Namely, we require that the predator saturation rate β(u) be proportional to the inherent growth rate r(u), and that the carrying capacity k(u) be inversely proportional to the inherent growth rate r(u). We analyze all equilibrium possibilities, both extinction and non-extinction cases. Furthermore, applying the Poincaré–Bendixson theorem, we show that all bounded orbits equilibrate for six basic cases; i.e. that either orbits are unbounded or they are bounded and approach an equilibrium point. Finally, we include two illustrative examples where we compare evolutionary cases to non-evolutionary cases. In these instances, we illustrate that evolution is beneficial and maximizes the population level.

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1. Introduction In 1981, Phillip Tuchinsky published a paper concerning spruce budworm outbreaks, a serious ecological problem that eastern Canada and the northeastern United States have faced since at around the early eighteenth century. Every 30-70 years, a cycle begins where the spruce budworm population in an infected area greatly increases over a short period of time and kills a majority of mature balsam fir trees, whose needles they consume.

In the first five years of the outbreak cycle, spruce budworm population increases from about five larvae per tree to 21,000 larvae per tree. By the tenth year, the trees begin to die, and by the 14th year, eighty percent of the mature balsam firs are killed. Following this rapid outbreak, the lack of a food supply drives the population down, leading to regeneration of the forest. Tuchinsky argued that such a cycle illustrated the resilience of the spruce budworm population, despite the lack of stability [2].

Such outbreaks have been detrimental to the economy of the area. The use of insecticides helps prevent outbreaks, but is costly and potentially environmentally harmful. Thus, ecologist C.S. Holling, along with mathematicians Donald Ludwig and Dixon Jones, sought to create a mathematical model that would suggest ways we could manage the ecosystem and minimize outbreaks. Initially, Holling created a computer simulation with 78 state variables to model the scenario. Ludwig, with the help of Jones and Holling, turned this computer model into a system of three equations with three variables and was able to capture the fundamental dynamics of spruce budworm outbreaks.

Researchers modified the basic logistic population growth model by adding the effects of parasitism and predation. The resulting differential equation has one or three equilibrium points, depending on the parameters. As parameters are altered, a full hysteresis loop of a budworm population can be observed.

To our knowledge, this famous model has not been studied in the context of evolutionary adaptation. Thus, our goal is to consider the classic spruce budworm model in the presence of Darwinian evolution. To do so, we apply evolutionary game theory methods to the original Ludwig-Jones-Holling model. We focus on understanding how the spruce budworm equilibria are affected by the presence of evolutionary change.

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2. The Ludwig-Jones-Holling Differential Equations model (The Classic Spruce Budworm Model) We begin with the logistic model:

𝑑𝑥𝑑𝑡 = 𝑟𝑥 1−

𝑥𝑘

(1) where x is budworm density, r is the inherent (i.e. density dependent) growth rate, and k is the carrying capacity. This equation represents birth and death rates, and their control by population density (e.g. by limited food supply).

Next, we consider the effects of both parasitism and predation on the budworm population. It is assumed that very low populations are hardly affected by predation and parasitism, and so the effect on population growth rate is negligible. Specifically, Ludwig [1] assumes that the rate of predation grows quadratically as a function of population density x for a small population, but is bounded and eventually approaches a saturation level, which we call β. A parameter α is selected to control the saturation rate. Following Ludwig, we take the rate of predation to be:

𝛽𝑥!

𝛼! + 𝑥!  . (2)

Combining the logistic and predation models gives the Ludwig-Jones-Holling differential equations model:

𝑑𝑥𝑑𝑡 = 𝑟𝑥 1−

𝑥𝑘 − 𝛽

𝑥!

𝛼! + 𝑥! (3)

This equation has the following equilibrium possibilities: 1.) x = 0, 2.) and x satisfying the equation: 𝑟 1− !

!= 𝛽 !

!!!!!.

The first equilibrium possibility is an extinction equilibrium. Since a negative budworm

density does not make sense, the stability of the extinction equilibrium can be found by considering perturbations where x is small and positive. For such x, we have x >> x2 and from

𝑑𝑥𝑑𝑡 = 𝑟𝑥 − 𝑥

! 𝑟𝑘 +

𝛽𝛼! + 𝛽!

(4)

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we see that !"!"> 0 since the x term dominates 𝑥! on the right side of the equation above. It

follows that the extinction equilibrium is unstable. Another way to show this is to consider the linearization of the function evaluated at x=0. We get

𝑑𝑥𝑑𝑡 ≈ 𝑟𝑥

(5) Since r > 0, the Linearization Principle implies x = 0 is unstable.

Other equilibria occur when the per capita growth rate of the budworm population (per unit of habitat, per unit of time) equals the per capita death rate due to predation.

𝑟 1−

𝑥𝑘 = 𝛽

𝑥𝛼! + 𝑥!

(6) Following Tuchinsky, we can graph the following functions

𝑓 𝑥 =  𝑟 1−

𝑥𝑘

(7a)

𝑔 𝑥 = 𝛽𝑥

𝛼! + 𝑥! (7b)

to see where they intersect in order to analyze the values of x that yield equilibrium values [2].

Depending on the values of the parameters, we can have one or three equilibrium points (see Figure 1). Tuchinsky explains the ecological significance of these equilibria [2]. In a young forest, the growth rate r is small and there is one equilibrium (Figure 2). Budworm population will be at or near the small stable equilibrium, which Tuchinsky calls the survival population, or xa.

As the forest grows, r increases and the carrying capacity k remains steady, resulting in three equilibria (Figure 3). The first equilibrium is still xa, the survival population. The additional equilibria are unstable (xb) and stable (xc).

As r increases more, the stable equilibrium xa and the unstable equilibrium xb move toward each other and disappear. Eventually, we reach the case in Figure 4, where there is one stable equilibrium xc. This represents an outbreak of the budworm population.

As the forest begins to die, we follow the figures in reverse order, returning to Figure 2. Thus, the Ludwig-Jones-Holling differential equations model results in a hysteresis loop for the dynamics of the budworm population.

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Figure 1. Equations (7a) and (7b) are plotted with two examples of f(x). Intersection values of f and g yield equilibrium values. We obtain one or three equilibrium points depending on the parameters.

Figure 2. The spruce budworm population in a young forest.

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Figure 3. The spruce budworm population with three equilibrium points.

Figure 4. The spruce budworm population during an outbreak.

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3. Evolutionary Game Theory (EGT) Version of the Spruce Budworm Model

The goal of this paper is to consider the classic spruce budworm model in the presence of Darwinian evolution. We will apply the original equation (3) above to the method of evolutionary game theory, as used in [3]. A model for the evolutionary dynamics of a logistically growing population is:

𝑢! =  𝜎!𝜕𝐺(𝑢, 𝑥)𝜕𝑢  ,

(8a) 𝑥! =  𝑥𝐺 𝑢, 𝑥 ,

(8b) 𝐺 𝑢, 𝑥 ≗  𝑟(𝑢) 1−

𝑥𝑘(𝑢)

(8c) where u is a mean of a phenotypic trait subject to Darwinian evolution. The mean phenotypic trait affects the parameters r and k. The constant coefficient 𝜎2 ≥ 0 is the variance of the trait and is a measure of the speed of evolution. In this context, the per capita growth rate G(u, x) is referred to as fitness. Equation (8a), called Breeder’s or Lande’s equation, postulates that change in the mean trait is proportional to the fitness gradient [5].

We consider the evolutionary game theory (EGT) version of the Ludwig-Jones-Holling equation, equation (3) above, namely (8a) and (8b) with

𝐺 𝑢, 𝑥 ≗  𝑟(𝑢) 1−𝑥

𝑘(𝑢) − 𝛽(𝑢)𝑥

α! + 𝑥!

(9)

This model assumes r, k, and 𝛽 depend on the mean trait u, but the parameter α does not. We will let c = α2, with c > 0. As in the original Ludwig-Jones-Hollings equation, k represents the carrying capacity, r the inherent growth rate, and β the saturation level of predation.

Mathematically, we assume the functions k(u), r(u), and β(u) are positive-valued and twice-continuously differentiable with non-zero second derivatives. We also assume that the critical points of these functions are isolated. The function G is the per capita growth rate and is a measure of the fitness. Thus, G is a function of the population density x and the mean trait u.

Biologically speaking, an increase in the growth rate of a population due to beneficial

evolution would coincide with the increase in saturation level of predation, due to an improved physical trait and a higher population. Similarly, a decrease in the growth rate of a population would coincide with a decrease in the saturation level of predation. This is mathematically expressed by the requirement that r(u) and β(u) have the same monotonicities, and thus, share the same critical points. Additionally, it is biologically reasonable to assume that the carrying capacity of a population would decrease as both the inherent growth rate and predation level increased. Motivated by these trade-offs, we require that β(u) to be a constant multiple of r(u) and that k(u) be inversely proportional to r(u). We let

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𝛽(𝑢) = 𝑝𝑟 𝑢 ,𝑝𝜖ℝ!. (10a)

𝑘(𝑢) = 𝑞1

𝑟 𝑢 , 𝑞𝜖ℝ!.

(10b) Applying equations (8a) and (8b), we get the following equations for G and the derivative of G

𝐺 𝑢, 𝑥 ≗  𝑟 𝑢 1−𝑥𝑟 𝑢𝑞 −

𝑝𝑥𝑐 + 𝑥!  

(11a)

𝜕𝐺(𝑢, 𝑥)𝜕𝑢 = 𝑟

! 𝑢 1−2𝑥𝑟 𝑢𝑞 −

𝑝𝑥𝑐 + 𝑥!

(11b) and we obtain the following plane-autonomous system of equations:

𝑢! =  𝜎!𝑟! 𝑢 1−2𝑥𝑟 𝑢𝑞 −

𝑝𝑥𝑐 + 𝑥!  ,

(12a)

𝑥! =  𝑥𝑟 𝑢 1−𝑥𝑟 𝑢𝑞 −

𝑝𝑥𝑐 + 𝑥! .

(12b) Note that the rates 𝑢′ and 𝑥′ in the revised system are dependent on the population density x and the inherent growth rate r(u).

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4. Equilibria and Local Stability The equilibrium equations for the EGT model (12) are

0 =𝜕𝐺(𝑢, 𝑥)𝜕𝑢  ,

(13a) 0 =  𝑥𝐺 𝑢, 𝑥

(13b) assuming 𝜎! > 0, or that evolution occurs. From these equations, we get the following equilibrium possibilities: 1.) (ue,xe) = (ur,0), is an equilibrium if and only if 𝑟′(𝑢!) = 0 2.) (ue,xe) = (ur, x*) is an equilibrium if and only if 𝑟′(𝑢!) = 0 and x* is an equilibrium to the original Ludwig-Jones-Holling model (equation 3). The first equilibrium possibility is an extinction equilibrium, i.e. one whose x component equals zero. This exists at and only at a mean trait ur, which is a critical point of r(u). The second equilibrium possibility occurs at a spruce budworm equilibrium, x*, and at ur, the same critical trait of r(u). To study the local stability of the equilibria, we use the Jacobian

𝐽 𝑢, 𝑥 =

𝜕𝑢!

𝜕𝑢𝜕𝑢!

𝜕𝑥𝜕𝑥!

𝜕𝑢𝜕𝑥!

𝜕𝑥

(14a) where

𝜕𝑢!

𝜕𝑢 =𝜕𝜕𝑢 𝜎

2 𝜕𝐺(𝑢, 𝑥)𝜕𝑢

(14b) 𝜕𝑢!

𝜕𝑥=𝜕𝜕𝑥 𝜎

! 𝜕𝐺(𝑢,𝑥)𝜕𝑢

(14c) 𝜕𝑥!

𝜕𝑢=𝜕𝜕𝑢 𝑥𝐺 𝑢,𝑥

(14d) 𝜕𝑥!

𝜕𝑥=𝜕𝜕𝑥 𝑥𝐺 𝑢,𝑥

(14e) The equilibrium equation (14d) makes the lower left entry of the Jacobian equal zero when evaluated at an equilibrium. Thus, the eigenvalues of the Jacobian are the diagonal entries evaluated at the equilibrium points.

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𝜆! = 𝜎! 𝑟!! 𝑢 1−𝑥𝑘 − 𝛽

!! 𝑢𝑥!

𝑐 + 𝑥!!,! !(!!,!!)

(15a)

𝜆! = 𝑟 𝑢 1−𝑥𝑘 − 𝛽

! 𝑢𝑥

𝑐 + 𝑥! −𝑟 𝑢𝑘 − 𝛽 𝑢

𝑐 − 𝑥!

𝑐 + 𝑥! !!,! !(!!,!!)

(15b) Evaluating equation (15a) and (15b) at the extinction equilibrium, (ur, 0), gives

𝜆! = 𝜎![𝑟!! 𝑢! ] (16a)

𝜆! = 1 (16b)

The following table summarizes the stability cases for the extinction equilibrium. Table 1. The Extinction Equilibrium (ur,0).

𝒓′′(𝒖𝒓) Stability > 0 Repeller (unstable node) < 0 Saddle

Evaluating equation (15a) and (15b) at the second equilibrium, (ur, x*), and simplifying gives

𝜆! = 𝜎! 𝑟!! 𝑢! 1−2𝑟 𝑢! 𝑥∗

𝑞 −𝑝𝑥∗

𝑐 + (𝑥∗)!

(17a)

𝜆! = 1−𝑟 𝑢! 𝑥∗

𝑞 −𝑝𝑥∗

𝑐 + (𝑥∗)!

(17b)

First, consider x* to be an unstable Spruce Budworm equilibrium [2]. Then, we know that 𝜆! is positive, and thus, we have an unstable equilibrium. Now, consider x* to be a stable spruce budworm equilibrium. We know that 𝜆! is negative. As a result, the component 1− !! !! !

∗

!− !!

∗

!!(!∗)! of 𝜆! must also be negative. Therefore, the sign of 𝜆! depends on

𝑟!! 𝑢! . Specifically, the sign of 𝜆! is opposite the sign of 𝑟!! 𝑢! . We use a table to summarize the possibilities. Table 2. The Non-Extinction Equilibrium (ur, x*).

1 −𝑟 𝑢! 𝑥∗

𝑞−

𝑝𝑥∗

𝑐 + (𝑥∗)!

𝑟!! 𝑢! 𝑟!! 𝑢! 1 −2𝑟 𝑢! 𝑥∗

𝑞−

𝑝𝑥∗

𝑐 + (𝑥∗)!

Stability

> 0 - - Unstable < 0 > 0 < 0 Stable node < 0 < 0 > 0 Saddle

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5. Global Analysis We consider the case when r(u) has only one critical point, ur. Biologically, this is an interesting case in that it assumes there is a unique trait ur, where r(u) is optimized. We obtain 6 different cases to be analyzed. In all of the following cases, we know the vertical phase plane lines located at a critical trait ur are orbits of Spruce Budworm equation (12b) with u=ur. First, consider that ur occurs at a minimum of r(u). There will either be one or three non-extinction equilibria. Since 𝑟!! 𝑢! > 0, we know that in the case of one equilibrium it is a stable node (Figure 5a). In the case of three equilibria, we know that the stable spruce budworm equilibria are stable nodes. However, the unstable spruce budworm equilibria may either be an unstable node (Figure 5b) or a saddle (Figure 5c).

(a) (b) (c) Figure 5.  The critical point occurs at a minimum of r(u). (a) One stable node. (b) Two stable nodes and one unstable node. (c) Two stable nodes and a saddle.

Next consider the case when ur occurs at a maximum of r(u). Again, there will either be one or three non-extinction equilibria. Since 𝑟!! 𝑢! < 0, we know that in the case of one equilibrium, it is a saddle (Figure 6a). In the case of three equilibria, we know that two of the critical points are saddles. However, the unstable spruce budworm equilibria may either be an unstable node (Figure 6b) or a saddle (Figure 6c).

(a) (b) (c) Figure 6. The critical point occurs at a maximum of r(u). (a) One saddle point. (b) Two saddles and one unstable node. (c) Three saddle points. For each of the six cases presented, we can apply the Poincaré–Bendixson theorem to show that all bounded orbits equilibrate. The Poincaré-Bendixson Theorem says that for S+, the forward limit set of an orbit bounded as t → +∞, either (a) S+ is an equilibrium (b) S+ is a limit cycle (c) S+ is a cycle chain [4].

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First we consider the three subcases where ur occurs at a minimum of r. In Figures 5a and 5b, there can be no limit cycle because one would have to cross a vertical orbit of the stable and unstable nodes. Also, since there is no saddle point, there can be no cycle chain. Thus, in either case, if the orbit is bounded, the critical point has to be a global attractor; otherwise, the orbit is unbounded. In Figure 5c, again no limit cycles exist because a vertical orbit would have to be crossed. Although there is a saddle point, there cannot be a saddle chain because there is no saddle orbit that returns to the saddle. Thus, bounded orbits at a minimum of r must approach an equilibrium point. Now we consider the three subcases where ur occurs at a maximum of r. In Figure 6a, we have a saddle point. There can be no limit cycle or cycle chain because either type of cycle must surround at least one equilibrium. In Figure 6b and 6c, there can be no limit cycles because one would again have to cross an orbit. Furthermore, there are no cycle chains because a cycle chain must surround at least one equilibrium. By the Poincaré–Bendixson theorem, we have shown that all bounded orbits for the six cases equilibrate, i.e. that either orbits are unbounded or they are bounded and approaching an equilibrium point.

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6. Examples This following examples illustrates the results we have presented in Sections 4 and 5. Suppose we let the inherent growth rate, r(u), be equal to a general quadratic equation,

𝑟 𝑢 = 𝑎 + !!𝑏𝑢!,

(18)

where a and b are real positive numbers. Then, we obtain the following equations for the carrying capacity and saturation level of predation,

𝛽 𝑢 = 𝑝(𝑎 + !!𝑏𝑢!),

(19a)

𝑘 𝑢 = 𝑞 !!!!!!!

! .

(19b)

Since the second derivative of r(u) is positive, the unique critical point 𝑢! = 0 occurs a minimum of r . For this example, we have the following system of equations

𝑢! =  𝜎!𝑏𝑢 1−2𝑥(𝑎 + 12 𝑏𝑢

!)𝑞 −

𝑝𝑥𝑐 + 𝑥!  ,

(20a)

𝑥! =  𝑥(𝑎 +12 𝑏𝑢

!) 1−𝑥(𝑎 + 12 𝑏𝑢

!)𝑞 −

𝑝𝑥𝑐 + 𝑥! .

(20b) In the Figure 7, we see that there is one non-extinction equilibrium, due to certain assigned numerical values. Thus, this example coincides with the scenario in Section 5, Figure 5a. The extinction equilibrium, located at the origin is a repeller, and the non-extinction equilibrium is a stable node. To compare the evolutionary case in Figure 7 to a non-evolutionary case, we turn off the variance by setting 𝜎! = 0. Again, we see one stable non-extinction equilibrium for different values of u in Figure 8.

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Figure 7. A phase plane for one non-extinction equilibrium with various trajectories shown. Parameter values are 𝜎! = 0.5, 𝑞 = 0.380, 𝑝 = 0.1, 𝑎 = 0.1, 𝑏 = 0.1, and 𝑐 = 1.

Figure 8. Phase trajectories for the non-evolutionary case with one non-extinction equilibrium. Parameter values are 𝜎! = 0, 𝑞 = 0.380, 𝑝 = 0.1, 𝑎 = 0.1, 𝑏 = 0.1, and 𝑐 = 1.

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Altering the numerical values of certain fixed parameters, we obtain the case where we have three non-extinction equilibria. Figure 9 coincides with the scenario in Section 5, Figure 5c. The extinction equilibrium, located at the origin is a repeller, and the non-extinction equilibria are two stable nodes with a saddle in between. Again, we compare the evolutionary case above to a non-evolutionary case by turning off the variance in Figure 10. Interestingly, we see one stable non-extinction equilibrium for larger values of u, and three non-extinction equilibria for smaller values of u.

Figure 9. A phase plane for three non-extinction equilibria with various trajectories shown. Parameter values are 𝜎! = 0.5, 𝑞 = 0.8, 𝑝 = 2, 𝑎 = 0.1, 𝑏 = 0.1, and 𝑐 = 1.

Figure 10. Phase trajectories for the non-evolutionary case with three non-extinction equilibria. Parameter values are 𝜎! = 0, 𝑞 = 0.8, 𝑝 = 2, 𝑎 = 0.1, 𝑏 = 0.1, and 𝑐 = 1.

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7. Discussion The plane-autonomous system of equations studied (12) is derived from the Ludwig-Jones-Holling differential equations model (3), as described in Section 2, and the application of evolutionary game theory, discussed in Section 3. The main assumption made in our analysis was the requirement that the predator saturation rate β(u) be proportional to the inherent population growth rate r(u) and that the predation free population carrying capacity k(u) (which is a surrogate for intra-specific competition) be inversely proportional to r(u). Essentially, we are imposing the assumptions that an increase in the growth rate of a population would coincide with the increase in saturation level of predation, due to an improved physical trait and a higher population. We are also assuming that the carrying capacity of a population would decrease as both the inherent growth rate and predation level increased. These relations are realistic biological possibilities.

We discuss the equilibria and local stability of the system of equations in Section 4. Two equilibrium possibilities are found, one being the extinction equilibrium. Table 1 and 2 illustrate the stability possibilities of each equilibrium.

In Section 5, a graphical global analysis is done under the assumption that there exists only one critical point, ur. We know that the vertical phase plane lines located at a critical trait ur are real orbits because they are Spruce Budworm equilibria. These were found by letting 𝜎! = 0, so that u is constant. However, we don’t know the geometry of the solutions of x, which depend on u. In Figure 5, we assume that ur occurs at a minimum of r. In each case, we see that the solutions equilibrate, so it must be bounded. In Figure 6, we assume that ur occurs at a maximum of r. However, we see that the orbits do not equilibrate, and so they are unbounded. These findings are a result of the application of the Poincaré–Bendixson theorem.

We apply our analysis to specific examples in order to get more insight in Section 6. In these examples, we are looking at a unique case where we made a specialized assumption for the value of the inherent growth rate (equation 18). Notice that in the non-evolutionary cases, we see small budworm population equilibria for mean phenotypic traits u of large magnitude. Without evolution, these species equilibrate to a low population level. Comparison of the evolutionary case in Figure 7 versus the non-evolutionary case in Figure 8 clearly illustrates that, in this example, evolution is beneficial because it maximizes the equilibrium population level.

We conjecture, in the evolutionary spruce budworm model, that the evolutionary model always approaches an equilibrium with a higher population component x than the non-evolutionary model. To support this conjecture, we point out that Figures 7 and 8 clearly support this. In addition, the more complicated example illustrated in Figures 9 and 10 also appears to support this idea. Mathematically, this is an open question that should be addressed.

In addition, it is not clear if the results we found would hold up without the specialized assumptions made in Equations 10a and 10b. It remains an interesting problem to formulate general theorems regarding the classic spruce budworm model in the presence of Darwinian evolution.

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References [1] D. Ludwig, D. D. Jones and C. S. Holling. Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest. Journal of Animal Ecology, Vol. 47, No. 1 (Feb., 1978), pp. 315-332. [2] P.M. Tuchinsky. Man in Competition with the Spruce Budworm. Birkhauser, October 1981. [3] T.L. Vincent and J.S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, 2005. [4] J.M. Cushing. Analysis of Ordinary Differential Equations, Math 355. Copyright 2013, J.M. Cushing. [5] J.S. Heywood. An Exact Form of the Breeder’s Equation for the Evolution of a Quantitative Trait Under Natural Selection. Evolution, Vol. 59, Issue 11 (Nov., 2005), pp. 2287-2298. [6] J.M. Cushing and J.T. Hudson. Evolutionary Dynamics and Strong Allee Effects. Journal of Biological Dynamics, 6(2), 2012, pp. 941-958. Acknowledgments I would like to thank Dr. Jim Cushing for being a dedicated thesis advisor. His knowledge and guidance were exceptional and made the process enjoyable. I would also like to thank the Mathematics Department and the Honors College for the ability to complete my Senior Honors Thesis.

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