Lotka-Volterra Model

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Lotka-Volterra Model

Introduction


The Lotka-Volterra equations are a pair of first order, nonlinear, differential equations used to describe the interaction dynamics between two species of organisms that have a predator-prey relationship. Despite the name, this set of differential equations was independently introduced two mathematicians, Alfred J. Lotka in 1925 and Vito Volterra in 1926.

Although these two mathematicians produced identical equations, the underlying assumptions behind these equations differed. Lotka’s version of the Lotka-Volterra Model assumed that as long as the population of species remain within the logical neighborhood of equilibrium point of the system, these populations of organisms will eventually become self-sustainable: reaching a constant state. However, since these neighbor regions were hard to estimate, and the parameters of his equations were not defined by system wide constants, the applicably of Lotka’s assumptions were greatly hampered. On the other hand, Volterra’s assumptions of the Lotka-Volterra Model resolved this issue by using parameters that were derived by global constants with defined biological meaning, these parameter generated could be generally applicable to estimate periodic and chaotic cycles of any populations that have prey-predator relationships.

Ever since the introduction of the Lotka-Volterra model in the mid 1920s, this set of differential equations has seen an increase in application by the biological world. Other than estimating the interactions dynamics between predator-prey, these equations were also used to describe other similar interaction relationships such as the host-parasitoid relationship as well. In response to the increase in applications, were modifications to the equations that made this set differential equations become more realistic by inserting limiting functions such as logistic growth functions and mass action functions.

The most significant impact of the Lotka-Volterra equation is that this was the first attempt by mathematicians and biologist to use seemingly simple mathematical equations to model the interaction dynamics between two species of organisms. Despite the simplicity in their underlying assumptions and estimation of parameters, the results estimated by this set of differential equations nevertheless correlated closely to real life observations of those species in their natural habitat.


The Model

Exponential Growth


The system of differential equations of the unmodified Lotka-Voltera model in their dimensionalized form,
 \frac{dN}{dt}=rN-cNP

 \frac{dP}{dt}=bNP-mP

characterize the simplest way to describe a predator-prey relationship. N represents the number of prey and P represents the number of predators.  \frac{dN}{dt} and  \frac{dP}{dt} represents the growth of each respective population against time. The differential equation for prey population change is equal to exponential prey growth minus the rate at which the predator captures the prey, i.e. Holling Type I functional response . The term rN, which characterizes the exponential prey growth, assumes that there is an unlimited supply of resources for the prey to utilize. The term cNP, also called the Holling Type I functional response, is considered a mass action term describing predator-prey encounters. It predicts a linear relationship between the prey and predator. Lastly, the differential equation for predator population is equal to the predator growth minus the rate of predator death.

In order to do a mathematical analysis of this system of equations the non-dimensionalized form of the equations can be used:

 \frac{dx}{dt}=r(1-y)x

 \frac{dy}{dt}= m(x-1)y

The steady states for this system of equations are found to be: (0,0) and (1,1) . The Jacobian matrix is:

 \left[ \begin{array}{cc}
r(1-y)& -rx \\ my & m(x-1) \end{array} \right]

Plugging in the steady states into the Jacobian allows one to determine the stability of the steady states. The Jacobians that result from plugging in the steady states (0,0) and (1,1) are:

J(0,0) = \begin{bmatrix}
0.5 & 0 \\
0 & -0.5 \\
\end{bmatrix}



Finding the eigenvalues from each Jacobian will allow us to determine the stability of the respective steady state. The eigenvalues for J(0, 0) are :\lambda_1 = 0.5,\quad \lambda_2 = -0.5.\,. The opposite signs of the eigenvalues inform us that at the steady state (0,0) we have a saddle point.

J(1,1) = \begin{bmatrix}
0 & -0.5 \\
0.5 & 0 \\
\end{bmatrix}



The eigenvalues for J(1,1) are :\lambda_1 = 0+ 0.5i,\quad \lambda_2 = 0-0.5i.\, . Since the real part of both eigenvalues is zero, it is determined that at (1, 1) the steady state is neutrally stable. The neutrally stable spiral is meant to represent the periodically oscillating prey-predator population. However, it is inadequate because of its structurally unstable center. Any slight deviation from this steady state and the system (population) goes into chaos; the system is stable only for specific conditions.

Overall, due to the fact that this system of equations assumes exponential growth for the prey, a Holling Type I functional response, and a neutrally stable steady state, this model is not a reliable or realistic characterization of prey-predator interactions. It must be altered in order to realistically depict the periodic oscillations of prey-predator populations.

Phase Plane for Exp. growth and Hollings Type 1. Source: Austin Ross Dike via pplane http://math.rice.edu/~dfield/dfpp.html

Modifications to the Model

Logistic Growth

Phase plane for the model, after incorporating logistic growth. Parameters beta=2, alpha=0.5.


The original Lotka-Volterra model assumes exponential growth of the prey population. This means that the prey population will continue to increase over time with the absence of predators. However, in natural populations, the prey population will not keep increasing in the absence of predators. The number of resources available in the area limits the populations. A given niche can only support a set number of individuals, called the carrying capacity. To make the model more realistic, a logistic growth was used in place of exponential growth. That is, the prey population levels reach a carrying capacity when predators are absent. The Lotka-Volterra model using logistic growth is shown below:

 \frac{dN}{dt}=rN(1-\frac{N}{K})-CNP

 \frac{dP}{dt}=bNP-mP

To find the steady states and stability of the Lotka-Volterra model with logistic growth of the prey population, the nondimensionalized form of the two equations are used:

 \frac{dx}{dt}=x(1-x-y)

 \frac{dy}{dt}=\beta (x-\alpha)y

The steady states are then (0,0), (1,0), and (α,1 − α), where α is between (0,1).

The Jacobian is given by the partial derivatives with respect to x and y at f(x) and g(x), where
f(x) = x(1 − xy)
g(x) = β(x − α)y
and it is

 \left[ \begin{array}{cc}
1-2x-y & -x \\
\beta y & \beta(x-\alpha) \end{array} \right]

For (0,0), the trace is 1 − βα and the determinant is − βα. Because the determinant is always negative, a saddle point exists at (0,0).

For (1,0), the trace is − 1 + β(1 − α) and the determinant is β(α − 1).
When α > 1, the determinant is less than zero, and the trace is less than zero, so (1,0) is a stable state.
When 0 < α < 1, the determinant is greater than zero, and (1,0) is a saddle point.

For (α,1 − α), the trace is − α and the determinant is αβ(1 − α).
Because 0 < α < 1, the determinant is always positive. However, the trace is always negative, so (α,1 − α) is stable.

Thus, with the logistic growth model, the Lotka-Volterra model loses its neutrally stable state and for 0 < α < 1, the population levels of prey and predators sinks to (α,1 − α), while for α > 1, the population level of prey goes to the carrying capacity, and the predator becomes extinct.

Functional Responses

The functional response refers to the relationship between prey and predator. It describes the relationship between the density of prey in a region and the number of prey consumed by the predators in the region (Wikipedia. "Functional Responses"). There are three main types of functional responses.

Three Functional Response Types.

Type I


The Type I functional response is the original assumed relationship between predator and prey interactions. Type I assumes a linear relationship between the number of prey in a region and the number of prey consumed. That is, the number of prey consumed by predators is directly proportional to the number of prey in a region. Under the original exponential growth Lotka-Volterra model, a neutrally stable state is found, while under the logistic growth Lotka-Volterra model, a stable node indicates the long term stability of the model.

This functional response is not very realistic because it is unlikely that predators will at larger population levels begin to kill and consume an infinite number of prey. While there may be a relationship at first, predators are limited by the time it takes to hunt and kill the prey, so at a certain prey population level, predators will likely not be consuming more prey with a spike in prey population levels. Holling devised two alternate functional responses to describe the rate at which the predator consumes prey in a region: Holling Type II and Holling Type III.

Holling Type II Functional Response


Model with Log. Growth and Hollings Type 2. Gamma<3
Model with Log. Growth and Hollings Type 2. Gamma>3
Bifurcation Diagram for our modified model. Notice the Hopf Bifurcation Point at gamma=3.

One way to improve the Lotka-Volterra Model is to include a more realistic functional response. The rate at which predators capture prey can be modeled as a hyperbolic function that saturates due to the limited time that predators have to search for prey. At some point, increases in prey density will have no effect on the amount of prey caught by predators. The following is a derivation that uses the time it takes to handle prey (Kot, 2001):

Let
T = total time
Th = handling time for each prey (time it takes to eat)
N = number of potential prey
V = number of prey caught

Assuming that the number of prey caught (V) is proportional to the number of potential prey (N) and time spent searching (T-VTh),
V = αN(TVTh) , where α is the proportionality constant.

If solved for V, the equation becomes
V = \frac{\alpha TN}{1+\alpha ThN}

This equation can be re-written into the Michaelis-Menten form as
\phi (N) = \frac{cN}{a+N}
where c = \frac{T}{Th} and a = \frac{1}{\alpha Th}

Incorporating the Holling Type II Functional Response into the logistic growth model yields
 \frac{dN}{dt}=rN(1-\frac{N}{K})-\frac{cN}{a+N}P

 \frac{dP}{dt}=\frac{bN}{a+N}P-mP

The non-dimensionalized forms of these equations are
\frac{dx}{dt} = x(1-\frac{x}{\gamma}) - \frac{xy}{1+x}

\frac{dy}{dt} = \beta (\frac{x}{1+x} - \alpha)y

The x-nullclines are:
x = 0 and y = (1+x)(1-\frac{x}{\gamma})
The y-nullclines are:
y = 0 and x = \frac{\alpha}{1-\alpha}
The steady states are:
(0,0), (γ,0), and (\frac{\alpha}{1-\alpha},(1+\frac{\alpha}{1-\alpha})(1-\frac{\alpha}{\gamma (1-\alpha)}))

The stability of the steady states can be summarized in a bifurcation diagram plotting x against γ.



The emergence of periodic solutions when γ = 3 suggests that the addition of the Holling Type II functional response into the logistic growth Lotka-Volterra model yields more biologically realistic solutions.

Holling Type III Functional Response

Holling Type III improves the modeled predator-prey dynamic in type II, and models the consumption rate, φ(t), as a sigmoid. Similar to the behavior exhibited in type II, the consumption rate φ(t) asymptotically approaches a steady state value for large prey density. At low prey density, the predatory consumption rate is modeled as exponential growth. This relationship is a function of predator learning time and prey switching. (Wikipedia. "Functional Responses"). A type III functional response takes the form

\phi(N) = \frac{cN^2}{a^2+N^2}
where φ is the consumption rate and N is the prey density . (Kot 2003).

Learning Time

At low prey density, the prey is scarce. Predators are unable to learn the eating, habitation, and other habits of the prey species. This inability to learn due to scarcity causes relatively inefficient predation, and the consumption rate is reduced. At higher prey densities, the prey is available in numbers abundant enough that the predator species is able to learn how to effectively hunt the prey. Predation becomes more efficient, and the consumption rate increases.

Prey Switching

Predator species that are not tied to the specific prey under consideration have incentive to switch to alternative prey sources if the modeled prey density is low. Since the prey is harder to find, predators switch to other, more abundant prey species, causing the consumption rate to drop. High prey densities remove the incentive for predators to switch to other prey species, causing the consumption rate to increase until reaching steady state.

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