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Mathematical model Lotka Volterra
2022-07-05 23:51:00 【jeff one】
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.
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