The Predator Prey Model

Analyzing the Lotka Volterra Predator-Prey Model Qualitatively:

The Lotka Volterra equations are a pair of 1st order non linear differential equations used to mathematically model a biological system in which a predator species and prey species interact.

The equations:
x’ =  x ( p – qy )
y’ = -y ( r – sx )

(x= prey population, y=predator population,     p, q, r, s = constants for a given system (>0) ,    x’ = dx/dt    , y’= dy/dt)

Assumptions:
i) prey finds ample food at all times
ii) food of predator depends entirely on prey
iii) rate of change of population is proportional to its size
iv) environment does not change in favour of any one species

Prey:
dx / dt   =  px – qxy
Reproduce exponentially unless subject to predation ,hence px term
Rate of predation is proportional to rate at which predator and prey meet, hence qxy term
So change in prey population = growth – rate at which preyed upon = dx / dt

Predator:
dy / dt = sxy – ry
Growth of population due to predator prey interaction represented by sxt (rate of growth of predator need not be equal to rate of decay of prey)
Exponential decay due to loss rate because of natural death in absence of prey, hence ry term
Change in popluation = growth due to predation – natural death = dy/ dt

I – Solving by method of Linearization:

Equilibrium points:   x’ = 0 , y’ = 0
So solution for these points:  x=0,  y=0   and x=r/s  , y=p/q

In the general case:
x = r/s + X    ,     y = p/q + Y
where X, Y are deviations of x, y from equilibrium

Replacing in the equation (i.e. translating r/s , p/q to the origin)

X = x – r/s
X, = x,=  x(p-qy)   = px – qxy
= p (r/s + X)  – q (r/s + x)(p/q + Y)
= -qr/s Y – qXY

Y= y- p/q
Y, = y, = -y(r-sx)   = sxy – ry
= s(r/s + X)(p/q + Y) – r(p/q +Y)
=ps/q X + sXY

i.e.:
dX/dt = -qr/s Y -qXY
dY/dt = sp/q X + sXY

assuming deviations are small- linearizing- neglecting XY term
dX/dt = -qr/sY    ,   dY/dt=sp/qX
so                     dY/dX = -ps2/q2r  *   X/Y
then                  q2rY dY  =  -ps2X dX
integrating
q2rYY  +  ps2 X2  = k2

which is the equation of an ellipse surrounding the origin in the XY plane with k=constantellipse 1So in xy plane, the non linear system has a centre at (p/q, r/s) surrounded by a closed trajectory so that there is a cyclic variation about the critical point:
ellipse 2

As we move along the trajectory from P to Q, the predator population sharply increases in number from a maximum of prey to a maximum of predator. Then, as the prey reach a minimum, the predator population also decreases. Now, from R to P, the predator first decreases and then increases as we go from a minimum of prey to a maximum of prey. And so , the cycle continues…

graph

The change in populations of both the predator and the prey population follows simple harmonic motion with a constant phase difference between the two populations.

II – Analyzing by Phase Space diagrams:
dx/dt = x(p-qy)            dy/dt = -y(r-sx)
eliminating t:    (p-qy)/y * dy   = -(r-sx)/x *dx
integrating:       p logy –  qy = -r logx + sx + logK

Yp  e-qy   =  K x-r  esx
(Let
K =x0r   y0p  e-dx0  – qy0
z = Yp  e-qy  
w = K x-r  esx
z = w   )

phase diagram

From the above diagram:
For maximum in yz plane i.e. A corresponding to M in wz plane,  A’ and A” in wx plane determine bounds for x in xy plane
For minimum in wx plane, i.e. B corresponding to N in wz plane, B’ and B” in yz plane determine bounds fory in xy plane
So points P1, p2, Q1, Q2 are determined for xy phase space
Other points are determined by moving along line MN in wz projecting to C1 over C3 and to C2 over C3

Changing the value of K raises or lowers B, expanding or contracting C
So different values of K gives us a family of ellipses around the centre S
For point S,   x =r/s ,   y= p/q
As time increases, an arbitrary point on C3 moves anticlockwise since when x < r/s  then dy/dt is -ve so the point moves along Q2 P1 Q1 and similarly for the other way around.

Sources:
‘Differential Equations’ by G F Simmons
‘Advanced Engineering Mathematics’ by E Kreyszig
physicsforums.com
wikipedia.org/wiki/Lotka_volterra
scholarpedia.org/article/Predator-prey_model

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