Consider the two species competition model given by d a d t = [ λ 1 a / ( a + K 1 ) ] − r a b ⋅ a b − d a , ( 1 ) d b d t = [ λ 2 b ∗ ( 1 − b / K 2 ) ] − r b a ⋅ a b , t &gt; 0 , ( 2 )for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here λ , λ 2 , K 1 , K 2 , r a b , r b a and d are all positive parameters. (a) Describe the biological meaning of each term in the two equations.=&gt; A series expansion of 1/(a+K1), gives 1 a + K 1 ≈ K 1 − a K 1 2 + O ( a 2 ) Now, d a d t = [ λ 1 a × a + K 1 K 1 2 ] − r a b a b − d a , λ1 a represents the exponential growth of populationda represents the exponential decay of population λ1 is the growth rated is the decay ratewhat does r_(ab) ab represent?The first term of RHS equation 1 : [ λ 1 a × a + K 1 K 1 2 ] represents logistic growth at a rate λ1 with carrying capacity K1. d b d t = [ λ 2 b × 1 − b K 2 ] − r b a a b , λ2 b represents the exponential growth of population λ2 is the growth ratewhat does r_(ba) ab represent?The first term of RHS equation 2 : [ λ 2 b × 1 − b K 2 ] represents logistic growth at a rate λ2 with carrying capacity K2.

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Consider the two species competition model given by            d      a              d      t        =  [            λ        1    a      /    (  a  +  K  1  )  ]  −      r          a      b        ⋅  a  b  −  d  a  ,                                (  1  )            d      b              d      t        =  [            λ        2    b  ∗  (  1  −  b      /    K  2  )  ]  −      r          b      a        ⋅  a  b  ,                       t  &amp;gt;  0  ,                          (  2  )for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here   λ  ,  λ  2  ,      K    1    ,      K    2    ,      r          a      b        ,      r          b      a       and d are all positive parameters. (a) Describe the biological meaning of each term in the two equations.=&amp;gt; A series expansion of 1/(a+K1), gives      1          a      +      K      1        ≈            K      1      −      a              K              1        2            +      O      (              a        2            )      Now,            d      a              d      t        =  [  λ  1  a  ×            a      +      K      1              K              1        2              ]  −      r          a      b        a  b  −  d  a  ,  λ1 a represents the exponential growth of populationda represents the exponential decay of population  λ1 is the growth rated is the decay ratewhat does r_(ab) ab represent?The first term of RHS equation   1  :  [  λ  1  a  ×            a      +      K      1              K              1        2              ] represents logistic growth at a rate   λ1 with carrying capacity K1.            d      b              d      t        =  [  λ  2  b  ×            1      −      b              K      2        ]  −      r          b      a        a  b  ,  λ2 b represents the exponential growth of population  λ2 is the growth ratewhat does r_(ba) ab represent?The first term of RHS equation   2  :  [  λ  2  b  ×      1    −    b        K    2    ] represents logistic growth at a rate   λ2 with carrying capacity K2.

Theresa Wade · Accepted Answer

A typical Interspecific competition between two species a and b can be represented by the following equations:dadt=λ1a[1−ak]This is the logistic population growth model in the absence of competing species b.Now ,dadt=λ1a[1−ak1−β12bk1]is the logistic population growth model for a under the presence of species b. Now we are going to add one more term to the model to represent the decay by exponential decay model.Exponential decay model isdadt=−λdaAdd this term to the original equation, we getdadt=λ1a[1−ak1−β12bk1]−λdaNow to define each of the term in the above model (ODE).λ1 = logistic population growth rate.k1 - Carrying capacity of species a**β12b - can be thought of as the decrease in growth rate of species &quot;a&quot; due to the presence of species &quot;b&quot;λd = exponential decay rate.Similarly,dbdt=λ2b[1−bk2−β21ak2]Now to define each of the term in the above model (ODE).λ2 = logistic population growth rate of b.k2 - Carrying capacity of species b**β21a - can be thought of as the decrease in growth rate of species &quot;b&quot; due to the presence of species &quot;a&quot;.This is typically the interspecific competition model. In your original equation, you have β12 to be rab

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