Improved model of a fishery: $dot N=rN(1-frac{N}{K})-Hfrac{N}{A+N}$












0












$begingroup$


Strogatz exercise $3.7.4.a:$



An improved model of a fishery is:



$$dot N=rNleft(1-frac{N}{K}right)-Hfrac{N}{A+N}.$$



a) Give a biological interpretation of the parameter $A$; what does it measure?



Here's what I did:



I non-dimensionalized the system $(frac{dN}{dt}=N(1-N)-hfrac{N}{a+N})$ and used the 'Manipulate' function of Mathematica to gain a better understanding of what '$a$' is biologically, but still, I couldn't see it.



So here's my question: How could I approach answering such a question and gain a better insight of it?



Thank you in advance.






ordinary-differential-equations dynamical-systems mathematical-modeling biology non-linear-dynamics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Have you tried Google?
    $endgroup$
    – John Douma
    Dec 24 '18 at 19:28










  • $begingroup$
    What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
    $endgroup$
    – Ross Millikan
    Dec 24 '18 at 19:33










  • $begingroup$
    @JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:34












  • $begingroup$
    @RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:37






  • 3




    $begingroup$
    @Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
    $endgroup$
    – Moo
    Dec 24 '18 at 19:45
















0












$begingroup$


Strogatz exercise $3.7.4.a:$



An improved model of a fishery is:



$$dot N=rNleft(1-frac{N}{K}right)-Hfrac{N}{A+N}.$$



a) Give a biological interpretation of the parameter $A$; what does it measure?



Here's what I did:



I non-dimensionalized the system $(frac{dN}{dt}=N(1-N)-hfrac{N}{a+N})$ and used the 'Manipulate' function of Mathematica to gain a better understanding of what '$a$' is biologically, but still, I couldn't see it.



So here's my question: How could I approach answering such a question and gain a better insight of it?



Thank you in advance.






ordinary-differential-equations dynamical-systems mathematical-modeling biology non-linear-dynamics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Have you tried Google?
    $endgroup$
    – John Douma
    Dec 24 '18 at 19:28










  • $begingroup$
    What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
    $endgroup$
    – Ross Millikan
    Dec 24 '18 at 19:33










  • $begingroup$
    @JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:34












  • $begingroup$
    @RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:37






  • 3




    $begingroup$
    @Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
    $endgroup$
    – Moo
    Dec 24 '18 at 19:45














0












0








0





$begingroup$


Strogatz exercise $3.7.4.a:$



An improved model of a fishery is:



$$dot N=rNleft(1-frac{N}{K}right)-Hfrac{N}{A+N}.$$



a) Give a biological interpretation of the parameter $A$; what does it measure?



Here's what I did:



I non-dimensionalized the system $(frac{dN}{dt}=N(1-N)-hfrac{N}{a+N})$ and used the 'Manipulate' function of Mathematica to gain a better understanding of what '$a$' is biologically, but still, I couldn't see it.



So here's my question: How could I approach answering such a question and gain a better insight of it?



Thank you in advance.






ordinary-differential-equations dynamical-systems mathematical-modeling biology non-linear-dynamics







share|cite|improve this question











$endgroup$




Strogatz exercise $3.7.4.a:$



An improved model of a fishery is:



$$dot N=rNleft(1-frac{N}{K}right)-Hfrac{N}{A+N}.$$



a) Give a biological interpretation of the parameter $A$; what does it measure?



Here's what I did:



I non-dimensionalized the system $(frac{dN}{dt}=N(1-N)-hfrac{N}{a+N})$ and used the 'Manipulate' function of Mathematica to gain a better understanding of what '$a$' is biologically, but still, I couldn't see it.



So here's my question: How could I approach answering such a question and gain a better insight of it?



Thank you in advance.






ordinary-differential-equations dynamical-systems mathematical-modeling biology non-linear-dynamics




ordinary-differential-equations dynamical-systems mathematical-modeling biology non-linear-dynamics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 24 '18 at 19:27









Namaste

1




1










asked Dec 24 '18 at 19:09









HeptapodHeptapod

716417




716417












  • $begingroup$
    Have you tried Google?
    $endgroup$
    – John Douma
    Dec 24 '18 at 19:28










  • $begingroup$
    What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
    $endgroup$
    – Ross Millikan
    Dec 24 '18 at 19:33










  • $begingroup$
    @JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:34












  • $begingroup$
    @RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:37






  • 3




    $begingroup$
    @Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
    $endgroup$
    – Moo
    Dec 24 '18 at 19:45


















  • $begingroup$
    Have you tried Google?
    $endgroup$
    – John Douma
    Dec 24 '18 at 19:28










  • $begingroup$
    What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
    $endgroup$
    – Ross Millikan
    Dec 24 '18 at 19:33










  • $begingroup$
    @JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:34












  • $begingroup$
    @RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
    $endgroup$
    – Heptapod
    Dec 24 '18 at 19:37






  • 3




    $begingroup$
    @Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
    $endgroup$
    – Moo
    Dec 24 '18 at 19:45
















$begingroup$
Have you tried Google?
$endgroup$
– John Douma
Dec 24 '18 at 19:28




$begingroup$
Have you tried Google?
$endgroup$
– John Douma
Dec 24 '18 at 19:28












$begingroup$
What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
$endgroup$
– Ross Millikan
Dec 24 '18 at 19:33




$begingroup$
What are the parameters? Presumably $N$ is the number of fish. What are $r,H,K? What is the model you are improving? The same without $A$? The interpretation of $A$ will depend on understanding the model.
$endgroup$
– Ross Millikan
Dec 24 '18 at 19:33












$begingroup$
@JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
$endgroup$
– Heptapod
Dec 24 '18 at 19:34






$begingroup$
@JohnDouma Yes, I even tried 'Mathematical Biology' by Murray. And also this article sciencedirect.com/science/article/pii/0198971582900011.
$endgroup$
– Heptapod
Dec 24 '18 at 19:34














$begingroup$
@RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
$endgroup$
– Heptapod
Dec 24 '18 at 19:37




$begingroup$
@RossMillikan Yes, $N$ is the number of fish, $r$ is how fast the population grows, $K$ is the carrying capacity, and $H$ is the harvesting rate. In the original model, $H$ is constant which creates a simple saddle-node bifurcation and it is very easy to comprehend.
$endgroup$
– Heptapod
Dec 24 '18 at 19:37




3




3




$begingroup$
@Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
$endgroup$
– Moo
Dec 24 '18 at 19:45




$begingroup$
@Heptapod: See number $3$: math.ucdavis.edu/~hunter/m207/set3_sol.pdf
$endgroup$
– Moo
Dec 24 '18 at 19:45










2 Answers
2






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votes


















2












$begingroup$

The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).



For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) approx (H/A)N$.



But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) nearrow H$ as $N to infty$.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of



    It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).



      For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) approx (H/A)N$.



      But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) nearrow H$ as $N to infty$.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).



        For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) approx (H/A)N$.



        But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) nearrow H$ as $N to infty$.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).



          For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) approx (H/A)N$.



          But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) nearrow H$ as $N to infty$.






          share|cite|improve this answer









          $endgroup$



          The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).



          For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) approx (H/A)N$.



          But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) nearrow H$ as $N to infty$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 24 '18 at 20:36









          Hans LundmarkHans Lundmark

          36.3k564116




          36.3k564116























              1












              $begingroup$

              I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of



              It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of



                It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of



                  It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.






                  share|cite|improve this answer









                  $endgroup$



                  I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of



                  It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 24 '18 at 19:44









                  Ross MillikanRoss Millikan

                  302k24201375




                  302k24201375






























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