Mathematical model Lotka Volterra

mathematical model Lotka-Volterra

origin ：

20 century 40 years ,Lotka（1925） and Volterra（1926） Laid a theoretical foundation for interspecific competition , The interspecific competition equation proposed by them has a significant impact on the development of modern ecological theory .

Content ：

Lotka-Volterra Model （Lotka-Volterra Interspecific competition model ） yes logistic Model （ Growth retardation model ） Extension of . Now set the following parameters ：

N1、N2： Are the population numbers of the two species
K1、K2： The environmental capacity of the two species
r1、r2 ： Are the population growth rates of the two species

According to the logistic model, there are the following relationships ：

among ：N/K It can be understood as the space that has been used （ be called “ Space item used ”）, be （1-N/K） It can be understood as unused space （ be called “ Unused space items ”）
When two species compete or use the same space ,“ Space item used ” You should also add N2 The occupation of space by the population . be ：

among ,α： species 2 For species 1 Competition coefficient , each N2 The space occupied by individuals is equivalent to α individual N1 Space occupied by individuals .

Then there are ,β： species 1 For species 2 Competition coefficient , each N1 The space occupied by individuals is equivalent to β individual N2 Space occupied by individuals . There is another ：

As we know ：

When species N1 population （ species 1） The environmental capacity of is K1 when ,N1 The inhibitory effect of each individual in the population on the growth of its own population is 1/K1; Empathy ,N2 The inhibitory effect of each individual in the population on the growth of its own population is 1/K2.

in addition , from （1）、（2） Two equations and α、β From the definition of ：

N2 Each individual in the population pairs N1 The effect of population is ：α/K1

N1 Each individual in the population pairs N2 The effect of population is ：β/K2

therefore , When species 2 Can inhibit species 1 when , It can be said that , species 2 For species 1 Influence > species 2 The impact on oneself , namely α/K1 > 1/K2.