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Lotka–Volterra equations
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==Dynamics of the system== In the model system, the predators thrive when prey is plentiful but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a [[population cycle]] of growth and decline. ===Population equilibrium=== Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0: <math display="block">x(\alpha - \beta y) = 0,</math> <math display="block">-y(\gamma - \delta x) = 0.</math> The above system of equations yields two solutions: <math display="block">\{y = 0,\ \ x = 0\}</math> and <math display="block">\left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}.</math> Hence, there are two equilibria. The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters ''α'', ''β'', ''γ'', and ''δ''. ===Stability of the fixed points=== The stability of the fixed point at the origin can be determined by performing a [[linearization]] using [[partial derivative]]s. The [[Jacobian matrix]] of the predator–prey model is <math display="block">J(x, y) = \begin{bmatrix} \alpha - \beta y & -\beta x \\ \delta y & \delta x - \gamma \end{bmatrix}.</math> and is known as the [[community matrix]]. ====First fixed point (extinction)==== When evaluated at the steady state of {{nowrap|(0, 0)}}, the Jacobian matrix {{mvar|J}} becomes <math display="block">J(0, 0) = \begin{bmatrix} \alpha & 0 \\ 0 & -\gamma \end{bmatrix}.</math> The [[eigenvalue]]s of this matrix are <math display="block">\lambda_1 = \alpha, \quad \lambda_2 = -\gamma.</math> In the model {{mvar|α}} and {{mvar|γ}} are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a [[saddle point]]. The instability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover. ====Second fixed point (oscillations)==== Evaluating ''J'' at the second fixed point leads to <math display="block">J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix} 0 & -\frac{\beta \gamma}{\delta} \\ \frac{\alpha \delta}{\beta} & 0 \end{bmatrix}.</math> The eigenvalues of this matrix are <math display="block">\lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}.</math> As the eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y)</math> on orbits. Thus orbits about the fixed point are closed and [[Dynamical system#Conjugation results|elliptic]], so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency <math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> and period <math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>. As illustrated in the circulating oscillations in the figure above, the level curves are closed [[orbit (dynamics)|orbit]]s surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without [[Damping ratio|damping]] around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>. The value of the [[constant of motion]] {{math|''V''}}, or, equivalently, {{math|1=''K'' = exp(−''V'')}}, <math>K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x}</math>, can be found for the closed orbits near the fixed point. Increasing {{math|''K''}} moves a closed orbit closer to the fixed point. The largest value of the constant {{math|''K''}} is obtained by solving the optimization problem <math display="block">y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max_{x,y>0}.</math> The maximal value of ''K'' is thus attained at the stationary (fixed) point <math>\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)</math> and amounts to <math display="block">K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math> where {{math|''e''}} is [[e (mathematical constant)|Euler's number]].
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