It is a calm afternoon on the Chitwan forest as deer listlessly wanders the green plain. Suddenly crouching Tiger starts to swoop in and snaps the treat for the whole family. Within no time, the population dynamics between Tiger and Deer, predator and prey, have shifted. Numerous such interactions happens each day, it is important to keep track of the trend, particularly if either population becomes threatened. For example: If count of Tigers increases, Deer may be endangered. On the other hand, too few deer, would trigger the starvation of all the Tiger. There has to be some kind of balance in order for both to survive.
There must be some models which can be used to simulate these kinds of predator-prey dynamics so that we can monitor various population. In the quest to answer the question of predator-prey population sustainability, and when are they doomed?. Lotka (in 1925) and Volterra (in 1926) invented equation that captures predator-prey dynamics and named it as Lotka-Volterra Equation.
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.
In economics Lotka Volterra equation has been used to simulate links between many if not all industries. It is done by modeling the dynamics of various industries by introducing trophic functions between various sectors, and ignoring smaller sectors by considering the interactions of only two industrial sectors.
In this section we limit our analysis within the predator-prey equations only. We formulate equation where the population of predator and prey change through time according to the pair of equations which are functions of x and y (f1 and f2):
Note that when we take x and y to the left hand side the equation exhibits growth of x and y:
In the given equations $\alpha$ is natural growth rate of x (Growth of Prey in absence of predator) and $\beta$ is coefficient of y. Negative sign of the coefficient explains that when there are more predators, the growth of prey falls.
Similarly $\gamma$ is natural growth rate of y (Growth of Predator in absence of Prey: Note that $\gamma$ has negative sign which denotes that in absence of Prey the Predator will die). Further $\delta$ is coefficient of x, the positive sign explains that when there are more Prey, the growth of Predator raises.
Now, next step is to determine steady state and linearize the non linear equation around it.
\[\begin{align} & \dot{x}=0 \\ & x(\alpha -\beta y)=0 \\ & \\ & \dot{y}=0 \\ & -y(\gamma -\delta x)=0 \\ \end{align}\]
Now Linearization around steady state
In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems.
Now for ease of calculation we individually solve for function (f1), first with respect to x and second with respect to y holding $\bar{x}$ and $\bar{y}$ constant
\[\begin{align}
& \frac{\partial {{f}_{1}}}{\partial x}=\alpha -\beta y \\
& \frac{\partial {{f}_{1}}}{\partial x}{{|}_{\bar{x}\bar{y}}}=\alpha -\beta \frac{\alpha }{\beta }=0 \\
& \\
& \frac{\partial {{f}_{1}}}{\partial y}=-\beta x \\
& \frac{\partial {{f}_{1}}}{\partial y}{{|}_{\bar{x}\bar{y}}}=-\beta \frac{\gamma }{\delta } \\
\end{align}\]
Similarly we solve for function (f2) with respect to x and second with respect to y holding $\bar{x}$ and $\bar{y}$ constant
\[\begin{align}
& \frac{\partial {{f}_{2}}}{\partial x}=\delta y \\
& \frac{\partial {{f}_{2}}}{\partial x}{{|}_{\bar{x}\bar{y}}}=\delta \frac{\alpha }{\beta } \\
& \\
& \frac{\partial {{f}_{2}}}{\partial y}=-\gamma +\delta x \\
& \frac{\partial {{f}_{2}}}{\partial y}{{|}_{\bar{x}\bar{y}}}=-\gamma +\delta \frac{\gamma }{\delta }=0 \\
\end{align}\]
Now substituting the results into linearization
\[\begin{align} & \dot{x}\approx 0+(-\frac{\beta \gamma }{\delta }(y-\bar{y})) \\ & =>\dot{x}\approx -\frac{\beta \gamma }{\delta }(y-\bar{y}) \\ & \\ & \dot{y}\approx \frac{\delta \alpha }{\beta }(x-\bar{x})+0 \\ & =>\dot{y}\approx \frac{\delta \alpha }{\beta }(x-\bar{x}) \\ \end{align}\]
Further we substitute the value of $\bar{x}$ and $\bar{y}$: \[\begin{align} & \dot{x}\approx -\frac{\beta \gamma }{\delta }y+\frac{\gamma \alpha }{\delta } \\ & \dot{y}\approx \frac{\delta \alpha }{\beta }x-\frac{\gamma \alpha }{\beta } \\ \end{align}\] The dynamic behavior of the change in X(Prey) and Y(Predator) depends on the roots of the above equation.The fast method for determining roots is by forming Characteristic equation. So in order to form the characteristic equation we transform the equation into matrix form. Let the matrix be named A
$\begin{align}
& \dot{y}=0 \\
& =>\frac{\delta \alpha }{\beta }-\frac{\gamma \alpha }{\beta }=0 \\
& =>x=\frac{\gamma }{\delta } \\
& \therefore x=\frac{\gamma }{\delta } \\
\end{align}$
The phase diagram of Lotka Volterra equations looks like the figure below. The directions of the arrow shows the movements in respective phases.
If we plot the the system of equations in excel with following parameters value we get spiral movements:
\[\begin{align} & \alpha =10 \\ & \beta =1 \\ & \gamma \text{= 20} \\ & \delta \text{=1} \\ \end{align}\]
As expected if we plot X (Pre) in x-axis and Y(Predator) in Y-axis, and we start just 10% above the steady state the system develops into outward moving spiral. (The time step used in calculation is 0.01)
I have constructed Lotka-Voltera dynamics in Excel and Mathematica, if you want to see the formation please click here.
Intuition: If natural birth rate of deer is 10%, Predator rate of consumption of prey is 1%, Predators death rate is 20% and Predators growth rate as a result of feeding on prey is 1%. Then the steady state value of Prey (X) is 20 and Predator (Y) is 10, if this is the starting point then the number of Prey and Predators can be maintained for ever (given all other things holding constant). Apart from the steady state intuition when we plot trend of X and Y against time we see fluctuating curve, there is clearly some periodic behavior going on here. This is exactly what one would expect from a predator-prey situation: the predator eat the prey, but then the predators starve from lack of food, and as the prey build up their numbers again, the predators begin to multiply. Although Lotka Volterra shows never ending cycle we can still observe the time frame the transitions takes to evolve.
There are criticism against Lotka-Volterra equation, however it has proved itself as strong base for development of several predator-prey equations and competition analysis.
There must be some models which can be used to simulate these kinds of predator-prey dynamics so that we can monitor various population. In the quest to answer the question of predator-prey population sustainability, and when are they doomed?. Lotka (in 1925) and Volterra (in 1926) invented equation that captures predator-prey dynamics and named it as Lotka-Volterra Equation.
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.
In economics Lotka Volterra equation has been used to simulate links between many if not all industries. It is done by modeling the dynamics of various industries by introducing trophic functions between various sectors, and ignoring smaller sectors by considering the interactions of only two industrial sectors.
In this section we limit our analysis within the predator-prey equations only. We formulate equation where the population of predator and prey change through time according to the pair of equations which are functions of x and y (f1 and f2):
$\begin{align}
& \dot{x}=x(\alpha -\beta y)={{f}_{1}}(x,y) \\
& \dot{y}=-y(\gamma -\delta x)={{f}_{2}}(x,y) \\
\end{align}$
\[\begin{align} & \frac{{\dot{x}}}{x}=(\alpha -\beta y)={{f}_{1}}(x,y) \\ & \frac{{\dot{y}}}{y}=-(\gamma -\delta x)={{f}_{2}}(x,y) \\ \end{align}\]
In the given equations $\alpha$ is natural growth rate of x (Growth of Prey in absence of predator) and $\beta$ is coefficient of y. Negative sign of the coefficient explains that when there are more predators, the growth of prey falls.
Similarly $\gamma$ is natural growth rate of y (Growth of Predator in absence of Prey: Note that $\gamma$ has negative sign which denotes that in absence of Prey the Predator will die). Further $\delta$ is coefficient of x, the positive sign explains that when there are more Prey, the growth of Predator raises.
Now, next step is to determine steady state and linearize the non linear equation around it.
Steady state:
We make $\dot{x}$ and $\dot{y}$ equal to 0
\[\begin{align} & \dot{x}=0 \\ & x(\alpha -\beta y)=0 \\ & \\ & \dot{y}=0 \\ & -y(\gamma -\delta x)=0 \\ \end{align}\]
Solving for steady state value of x and y gives us four results. We consider non-zero only :
\[\begin{align}
& \bar{x}=0,\bar{y}=0 \\
& \bar{x}=\frac{\gamma }{\delta },\bar{y}=\frac{\alpha }{\beta } \\
\end{align}\]
Now Linearization around steady state
$\begin{align}
& \dot{x}\approx \frac{\partial {{f}_{1}}}{\partial x}{{|}_{\bar{x}\bar{y}}}(x-\bar{x})+\frac{\partial {{f}_{1}}}{\partial y}{{|}_{\bar{x}\bar{y}}}(y-\bar{y}) \\
& \dot{y}\approx \frac{\partial {{f}_{2}}}{\partial x}{{|}_{\bar{x}\bar{y}}}(x-\bar{x})+\frac{\partial {{f}_{2}}}{\partial y}{{|}_{\bar{x}\bar{y}}}(y-\bar{y}) \\
\end{align}$
\[\begin{align} & \dot{x}\approx 0+(-\frac{\beta \gamma }{\delta }(y-\bar{y})) \\ & =>\dot{x}\approx -\frac{\beta \gamma }{\delta }(y-\bar{y}) \\ & \\ & \dot{y}\approx \frac{\delta \alpha }{\beta }(x-\bar{x})+0 \\ & =>\dot{y}\approx \frac{\delta \alpha }{\beta }(x-\bar{x}) \\ \end{align}\]
Further we substitute the value of $\bar{x}$ and $\bar{y}$: \[\begin{align} & \dot{x}\approx -\frac{\beta \gamma }{\delta }y+\frac{\gamma \alpha }{\delta } \\ & \dot{y}\approx \frac{\delta \alpha }{\beta }x-\frac{\gamma \alpha }{\beta } \\ \end{align}\] The dynamic behavior of the change in X(Prey) and Y(Predator) depends on the roots of the above equation.The fast method for determining roots is by forming Characteristic equation. So in order to form the characteristic equation we transform the equation into matrix form. Let the matrix be named A
$\left( \frac{{\dot{x}}}{{\dot{y}}} \right)=\left( \begin{matrix}
0 & \frac{-\beta }{\delta } \\
\frac{\delta \alpha }{\beta } & 0 \\
\end{matrix} \right)$
We determine trace and determinant of the above matrix which is then used in characteristic equation to know the nature of the roots.
(In linear algebra trace of n by n matrix is sum of diagonal from upper left to lower right.)
trace: tr(A)= 0
Determinant: |A|= $\alpha$ $\gamma$ > 0
We know the formula for Characterstic equation is ${{\text{R}}^{2}}+tr(A)+|A|$=0
So after substituting the value of Trace and Determinant the equation becomes ${{\text{R}}^{2}}+\alpha \gamma =0$
It is evident from the equation that there is 'No Real Roots' and trace of A is equal to zero. This means the dynamics will be in cycle.
This is further illustrated by drawing phase by drawing isoline for x and y
Isoline for x where $\dot{x}$ is zero, or in other words change in x is zero (x constant)
$\begin{align} & \dot{x}=0 \\ & \frac{-\gamma }{\beta }y+\frac{\gamma }{\delta }\alpha =0 \\ & \frac{\gamma }{\beta }(-y+\alpha )=0 \\ & -y+\alpha =0 \\ & \therefore y=\frac{\alpha }{\beta } \\ \end{align}$
Similarly we calculate:
So after substituting the value of Trace and Determinant the equation becomes ${{\text{R}}^{2}}+\alpha \gamma =0$
It is evident from the equation that there is 'No Real Roots' and trace of A is equal to zero. This means the dynamics will be in cycle.
This is further illustrated by drawing phase by drawing isoline for x and y
Isoline for x where $\dot{x}$ is zero, or in other words change in x is zero (x constant)
$\begin{align} & \dot{x}=0 \\ & \frac{-\gamma }{\beta }y+\frac{\gamma }{\delta }\alpha =0 \\ & \frac{\gamma }{\beta }(-y+\alpha )=0 \\ & -y+\alpha =0 \\ & \therefore y=\frac{\alpha }{\beta } \\ \end{align}$
Similarly we calculate:
The phase diagram of Lotka Volterra equations looks like the figure below. The directions of the arrow shows the movements in respective phases.
\[\begin{align} & \alpha =10 \\ & \beta =1 \\ & \gamma \text{= 20} \\ & \delta \text{=1} \\ \end{align}\]
As expected if we plot X (Pre) in x-axis and Y(Predator) in Y-axis, and we start just 10% above the steady state the system develops into outward moving spiral. (The time step used in calculation is 0.01)
Intuition: If natural birth rate of deer is 10%, Predator rate of consumption of prey is 1%, Predators death rate is 20% and Predators growth rate as a result of feeding on prey is 1%. Then the steady state value of Prey (X) is 20 and Predator (Y) is 10, if this is the starting point then the number of Prey and Predators can be maintained for ever (given all other things holding constant). Apart from the steady state intuition when we plot trend of X and Y against time we see fluctuating curve, there is clearly some periodic behavior going on here. This is exactly what one would expect from a predator-prey situation: the predator eat the prey, but then the predators starve from lack of food, and as the prey build up their numbers again, the predators begin to multiply. Although Lotka Volterra shows never ending cycle we can still observe the time frame the transitions takes to evolve.
There are criticism against Lotka-Volterra equation, however it has proved itself as strong base for development of several predator-prey equations and competition analysis.
