differential equation of a population growth and changeI want to formulate a system of equations and initial conditions of the following data: Each year the population1 grows by 4% and population2 by 2%. Also each year 3% of population1 leaves it and go to population2 and 1% vice versa, leaves population2 and goes to population1.The initial populations are: p o p u l a t i o n 1 = 20000000 , p o p u l a t i o n 2 = 5000000.I thought of the next formalism of this: x ( t + 1 ) = 1.04 ∗ x ( t ) − 0.03 ∗ x ( t ) + 0.01 ∗ y ( t ) y ( t + 1 ) = 1.02 ∗ y ( t ) − 0.01 ∗ y ( t ) + 0.03 ∗ x ( t )where x(t) is the size of population1 and y(t) is the size of population2. Is this is correct in those term or maybe I should reformulate it in another way ?I think that I need some how to get something like this (ignore the concrete number and function): X ′ ¯ = ( 1 1 − 2 − 2 ) X ¯ + ( e − t − 2 e − t ) with an initial conditions like that: X ¯ ( 0 ) = ( − 1 0 ) but I don't know how to do it, so will be grateful for some help.An idea I had is: x ′ ( t ) = x ( t + 1 ) − x ( t ) ( t + 1 − t ) = ( 1.04 − 0.03 − 1 ) x ( t ) + 0.01 y ( t ) is it correct ?

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differential equation of a population growth and changeI want to formulate a system of equations and initial conditions of the following data: Each year the population1 grows by 4% and population2 by 2%. Also each year 3% of population1 leaves it and go to population2 and 1% vice versa, leaves population2 and goes to population1.The initial populations are:   p  o  p  u  l  a  t  i  o  n  1  =  20000000  ,  p  o  p  u  l  a  t  i  o  n  2  =  5000000.I thought of the next formalism of this:  x  (  t  +  1  )  =  1.04  ∗  x  (  t  )  −  0.03  ∗  x  (  t  )  +  0.01  ∗  y  (  t  )  y  (  t  +  1  )  =  1.02  ∗  y  (  t  )  −  0.01  ∗  y  (  t  )  +  0.03  ∗  x  (  t  )where x(t) is the size of population1 and y(t) is the size of population2. Is this is correct in those term or maybe I should reformulate it in another way ?I think that I need some how to get something like this (ignore the concrete number and function):            X      ′        ¯    =      (                            1                          1                                      −          2                          −          2                      )        X    ¯    +      (                                        e                          −              t                                                            −          2                      e                          −              t                                            )  with an initial conditions like that:      X    ¯    (  0  )  =      (                            −          1                                      0                      )  but I don&#039;t know how to do it, so will be grateful for some help.An idea I had is:      x    ′        (    t    )    =            x              (        t        +        1        )            −      x              (        t        )                    (      t      +      1      −      t      )        =      (    1.04    −    0.03    −    1    )    x      (    t    )    +  0.01  y      (    t    )  is it correct ?

Alannah Yang · Accepted Answer

Step 1For population denoted by x, you get      x    ′    =  .04  x  −  .03  x  +  .01  yand for the second population denoted by y you get      y    ′    =  .02  y  +  .03  x  −  .01  yStep 2That simplifies to      x    ′    =  .01  x  +  .01  yand      y    ′    =  .03  x  +  .01  yNow you write it in matrix form.

bergvolk0k · Accepted Answer

Step 1For a year you&#039;re modeling the overall change in the first population,   x  (  t  +  1  )  −  x  (  t  ), as being the result of:- Some sort of inherent growth, adding   0.04  ×  x  (  t  ).- A loss to population 2, removing   0.03  ×  x  (  t  ).- New additions coming from population 2, adding   0.01  ×  y  (  t  ).That gives you  x  (  t  +  1  )  −  x  (  t  )  =  (  0.04  −  0.03  )    x  (  t  )  +  0.01    y  (  t  )  .Same sort of thing for population 2:  y  (  t  +  1  )  −  y  (  t  )  =  0.03    x  (  t  )  +  (  0.02  −  0.01  )    y  (  t  )  .Maybe easier to write                    x        (        t        +        Δ        t        )        −        x        (        t        )        )                            =        (        A        Δ        t        )        x        (        t        )        +        (        B        Δ        t        )        y        (        t        )                            y        (        t        +        Δ        t        )        −        y        (        t        )        )                            =        (        C        Δ        t        )        x        (        t        )        +        (        D        Δ        t        )        y        (        t        )            where A, B, C, D are rates since it looks like you might have an eye on shifting to a continuous model (  Δ  t  →  0). In that case you get the system                              x          ′                (        t        )                            =        A        x        (        t        )        +        B        y        (        t        )                                      y          ′                (        t        )                            =        C        x        (        t        )        +        D        y        (        t        )        ,            and typically you&#039;d have initial conditions.Step 2But, if that&#039;s not what you&#039;re after, the original problem would end up looking like      [                            x          (          t          +          1          )                                      y          (          t          +          1          )                      ]    =      [                            0.01                          0.01                                      0.03                          0.01                      ]        [                            x          (          t          )                                      y          (          t          )                      ]    ,and if you&#039;re working from initial conditions you end up with      [                            x          (          n          )                                      y          (          n          )                      ]    =            [                                    0.01                                0.01                                                0.03                                0.01                              ]        n        [                            x          (          0          )                                      y          (          0          )                      ]    .

Each year the population1 grows by 4% and population2 by 2%. Also each year 3% of population1 leaves it and go to population2 and 1% vice versa, leaves population2 and goes to population1.

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