Consider the single species population model defined by d R d t = g R R + R m − d R ,for t&gt;0, where g , R m , and d are all positive parameters and R ( 0 ) = R 0 (a) Describe the biological meaning of each term in the equation.(b) Determine the steady-states of the system and discuss any constraints on the model parameters for the model to admit biologically meaningful solutions.(c) Determine the steady-state stability and discuss any variation in this with respect to the model parameter values.a) gR represents the exponential growth of population, dR represents the exponential decay of population, g is the growth rate, and d is the decay rate. What do R and R m represent? How can I define the term g R R + R m ?What is R + R m ? Does it affect the gR for the grow?b) In single steady-state system, d R d t = 0. g R R + R m − d R = 0 g R − d R ( R + R m ) = 0 R ( g − d ( R + R m ) ) = 0either R=0 or g − d ( R + R m ) = 0 g − d R m = 0 ( R = 0 ) R m = g d R ∗ = g d ( R m = R ∗ )So, we have R 1 a s t = 0, R 2 ∗ = g d Are these correct? I am not sure constraint on the model parameters to admit biologically meaning solutions?c) to determine steady-state stability let f ( R ) = g R R + R m − d R d f d R = g l n ( R + R m ) − d .My differentiation may be wrong and don't know the term R m while differentiating with respect to R. I really don't know after that. and I know my answer is still incomplete.

Question

Consider the single species population model defined by            d      R              d      t        =            g      R              R      +              R        m              −  d  R  ,for t&amp;gt;0, where   g  ,      R    m  , and d are all positive parameters and   R  (  0  )  =      R    0  (a) Describe the biological meaning of each term in the equation.(b) Determine the steady-states of the system and discuss any constraints on the model parameters for the model to admit biologically meaningful solutions.(c) Determine the steady-state stability and discuss any variation in this with respect to the model parameter values.a) gR represents the exponential growth of population, dR represents the exponential decay of population, g is the growth rate, and d is the decay rate. What do R and       R    m   represent? How can I define the term            g      R              R      +              R        m              ?What is   R  +      R    m  ? Does it affect the gR for the grow?b) In single steady-state system,            d      R              d      t        =  0.            g      R              R      +              R        m              −  d  R  =  0  g  R  −  d  R  (  R  +      R    m    )  =  0  R  (  g  −  d  (  R  +      R    m    )  )  =  0either R=0 or   g  −  d  (  R  +      R    m    )  =  0  g  −  d      R    m    =  0           (  R  =  0  )      R    m    =      g    d        R          ∗        =      g    d             (      R    m    =      R          ∗        )So, we have       R    1          a      s      t        =  0,      R    2          ∗        =      g    d    Are these correct? I am not sure constraint on the model parameters to admit biologically meaning solutions?c) to determine steady-state stability let  f  (  R  )  =            g      R              R      +              R        m              −  d  R            d      f              d      R        =  g  l  n  (  R  +      R    m    )  −  d  .My differentiation may be wrong and don&#039;t know the term       R    m   while differentiating with respect to R. I really don&#039;t know after that. and I know my answer is still incomplete.

London Maldonado · Accepted Answer

R&amp;nbsp;m&amp;nbsp;is a constant. Your calculated steady states don&#039;t appear to be accurate. solving for R=0 is one steady state,&amp;nbsp;g&amp;nbsp;−&amp;nbsp;d&amp;nbsp;(&amp;nbsp;R&amp;nbsp;+&amp;nbsp;R&amp;nbsp;m&amp;nbsp;)&amp;nbsp;=&amp;nbsp;0&amp;nbsp;for R will give you the other. When computing&amp;nbsp;d&amp;nbsp;f&amp;nbsp;d&amp;nbsp;R&amp;nbsp;, you should use the quotient rule. Again,&amp;nbsp;R&amp;nbsp;m&amp;nbsp;is a constant so its derivative is zero.

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New Questions in Algebra I