<>Predator-Prey Model Predator and prey model

This is a very classic model in ecology

Suppose there are two species in an ecosystem , One of them is a herbivore , They constitute predators and prey respectively .

Take rabbits and foxes for example ：

Introduce variables ：

x ( t ) x(t) x(t) : Number of foxes

y ( t ) y(t) y(t)： Number of rabbits

If there were no rabbits , The number of foxes will decrease due to lack of food

d x d t = − a x , a > 0 \frac{\mathrm{d}x}{\mathrm{d}t}=-ax,a>0 dtdx​=−ax,a>0

in fact , There is an interactive relationship between rabbits and foxes in the ecosystem , The number of rabbits will decrease as the number of foxes increases , The number of foxes will also decrease as the number of rabbits decreases , Both affect each other from beginning to end . We use the product proportional to the number of the two to express this interaction ,
So a more accurate model can be written like this

d x d t = − a x + b x y (1) \frac{\mathrm{d}x}{\mathrm{d}t}=-ax+bxy\tag{1} dtd
x​=−ax+bxy(1)

Now consider the number of rabbits , If there were no Fox , And assume natural resources , Ample space , Then the rabbit will grow exponentially

d y d t = d y , d > 0 \frac{\mathrm{d}y}{\mathrm{d}t}=dy, d>0 dtdy​=dy,d>0

in fact , The number of rabbits will decrease with the increase of the number of foxes , This reduction is reflected in the interaction between the two organisms

\$KaTeX parse error: \tag works only in display equations

combination （1） and （2）, We can get a system of differential equations ：

d x d t = − a x + b x y \frac{\mathrm{d}x}{\mathrm{d}t}=-ax+bxy dtdx​=−ax+bxy

d y d t = d y − c x y \frac{\mathrm{d}y}{\mathrm{d}t}=dy-cxy dtdy​=dy−cxy

a , b , c , d a,b,c,d a,b,c,d Are constants
Their images are very interesting ：

This famous set of equations is called Lotka-Volterra predator-prey model
. In ecosystem , There is not only predation between species , And competition , The following model takes into account the competitive relationship between species

<>Competition Models Species competition model

Two organisms in the ecosystem compete for common resources , Like food , Living space, etc , This relationship is called competition . Now consider two species , For every species , The lack of each other's quantity will lead to the increase of their own quantity

Introduce variables ：

x ( t ) x(t) x(t) : species x

y ( t ) y(t) y(t)： species y

d x d t = a x , d y d t = c y \frac{\mathrm{d}x}{\mathrm{d}t}=ax,
\frac{\mathrm{d}y}{\mathrm{d}t}=cydtdx​=ax,dtdy​=cy

in fact , The competition among species will lead to a dynamic trend of change

d x d t = a x − b y \frac{d x}{d t}=a x-b y dtdx​=ax−by

d y d t = c y − d x \frac{d y}{d t}=c y-d x dtdy​=cy−dx

a , b , c , d a,b,c,d a,b,c,d Are constants

If we consider the interaction between species , To better describe the model , We use the product proportional to the number of the two to express this interaction

d x d t = a x − b x y \frac{d x}{d t}=a x-bxy dtdx​=ax−bxy

d y d t = c y − d x y \frac{d y}{d t}=c y-d xy dtdy​=cy−dxy

The following is a more accurate model , Specifically, it is a nonlinear system , Considering the species logistics Grow in a new way

d x d t = a 1 x − b 1 x 2 − c 1 x y \frac{d x}{d t}=a_{1} x-b_{1} x^{2}-c_1xy
dtdx​=a1​x−b1​x2−c1​xy

d y d t = a 2 y − b 2 y 2 − c 2 x y \frac{d y}{d t}=a_{2} y-b_{2} y^{2}-c_2xy
dtdy​=a2​y−b2​y2−c2​xy

We can get it by intuition alone , The species competition model is a life and death model , There should be a trend of change ：

Technology