How to prove the asymptotic stability of the trivial equilibrium of this system?I was trying to prove the asymptotic stability of the trivial equilibrium ( 0 , 0 ) of the two-dimensional non linear ODE system: d H d t = μ ( H + F ) 2 K 2 + ( H + F ) 2 − d 1 H − H ( σ 1 − σ 2 F H + F ) d F d t = H ( σ 1 − σ 2 F H + F ) − ( p + d 2 ) F where H and F are dependent variable and positive.All other parameters are non negative with σ 2 &gt; σ 1 .

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How to prove the asymptotic stability of the trivial equilibrium of this system?I was trying to prove the asymptotic stability of the trivial equilibrium   (  0  ,  0  ) of the two-dimensional non linear ODE system:                                          d            H                                d            t                          =        μ                              (            H            +            F                          )              2                                                          K              2                        +            (            H            +            F                          )              2                                      −                  d          1                H        −        H                  (                      σ            1                    −                      σ            2                                F                          H              +              F                                )                                                          d            F                                d            t                          =        H                  (                      σ            1                    −                      σ            2                                F                          H              +              F                                )                −        (        p        +                  d          2                )        F            where   H and   F are dependent variable and positive.All other parameters are non negative with       σ    2    &amp;gt;      σ    1  .

Jayvion Tyler · Accepted Answer

You need   K to be strictly positive. Because if   K  =  0 and   μ  &amp;gt;  0, then in a neighborhood of the origin,   d  H  /  d  t  ≈  μ  &amp;gt;  0. So, suppose   K  &amp;gt;  0.In the first equation, the terms   μ            (      H      +      F              )        2                            K        2            +      (      H      +      F              )        2             and   H      σ    2        F          H      +      F       are of second order of smallness near the origin. Thus, the sign of   d  H  /  d  t is determined by   −  (      d    1    +      σ    1    )  H which of course suggests stability.In the second equation,   −  H      σ    2        F          H      +      F       is of second order. Neglecting it, we are left with   H      σ    1    −  (  p  +      d    2    )  F. This looks troublesome, but if   H goes to zero,   F will be forced to follow.Let&#039;s summarize. For every   ϵ  &amp;gt;  0 there is a neighborhood of the origin in which                                          d            H                                d            t                                              &amp;lt;        −        (                  d          1                +                  σ          1                )        H        +        ϵ        (        H        +        F        )                                                  d            F                                d            t                                              &amp;lt;        H                  σ          1                −        (        p        +                  d          2                )        F        +        ϵ        (        H        +        F        )            (I work in the positive quadrant   H  ,  F  &amp;gt;  0, which is what you are interested in). Hence,            d      (      2      H      +      F      )              d      t        &amp;lt;  −  (  2      d    1    +      σ    1    )  H  −  (  p  +      d    2    )  F  +  3  ϵ  (  H  +  F  )which is negative, provided   2      d    1    +      σ    1    &amp;gt;  0,   p  +      d    2    &amp;gt;  0, and   ϵ is chosen sufficiently small.I took   2  H  +  F instead of   H  +  F so that the coefficient of   H on the right would have       σ    1  , increasing the chance of the coefficient being negative.

How to prove the asymptotic stability of the trivial equilibrium of this system? I was trying to pr

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