Trying to model red and grey squirrels

In summary: It's not fundamentally discrete, but continuous. You need to figure out what aG is as a function of time.
  • #1
LETS_GO
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0
TL;DR Summary
Hello, we are trying to change the max reproductive rate of grey squirrels. with a value starting at 1.2 and want it to go down to 0.3 with a step of 0.05.
Code: 
ClearAll["Global`*"]
(*R = reds, G = greys*)
(*S = susceptible, I = infected, R = recovered*)

tseries = {t, 0, 3};
vars = {HG[t], HR[t], SG[t], IG[t], RG[t], SR[t], IR[t], aG[t], qG[t]};

b = 0.4;    (*natural mortality rate, both species*)
\[Beta] = 0.7;    (*rate of virus transmission*)

aR = 1;       (*Reds max. reproductive rate*)
\[Alpha] = 26;       (*Reds mortaility rate due to virus*)
cR = 0.61;(*Reds competative effect on greys*)
KR = 60;   (*Reds carrying capacity, 5x5 km*)

\[Gamma] = 13;      (*Greys recovery rate from virus*)
cG = 1.65;(*Greys competative effect on reds*)
KG = 80;   (*Greys carrying capacity, 5x5 km*)

qR = (aR - b)/KR;  (*Reds crowding susceptibility*)
  (*Greys crowding susceptibility*)

(*initial conditions*)
SGinit = 10;
IGinit = 2;
RGinit = 0;
SRinit = 60;         
IRinit = 0;
HGinit = SGinit + IGinit + RGinit;
HRinit = SRinit + IRinit;

 eqns =
    (*total populations*)
  {HG[t] == SG[t] + IG[t] + RG[t],
   HR[t] == SR[t] + IR[t],
 
   aG[t] == aG[t - 1] - 0.05,(*aG = Greys max. reproductive rate -
   possible birth control??*)
   qG[t] == (aG[t] - b)/KG,  (*Greys crowding susceptibility*)
   (*SIR growth rates*)
 
   SG'[t] == ((aG[t] - (qG[t]*(HG[t] + (cR*HR[t]))))*HG[t]) - (b*
       SG[t]) - (\[Beta]*SG[t]*(IG[t] + IR[t])),
   IG'[t] == (\[Beta] *SG[t]*(IG[t] + IR[t])) - (b*IG[t]) - (\[Gamma]*
       IG[t]),
   RG'[t] == (\[Gamma]*IG[t]) - (b*RG[t]),
   SR'[t] == ((aR - (qR*(HR[t] + (cG*HG[t]))))*HR[t]) - (b*
       SR[t]) - ((\[Beta]*SR[t])*(IR[t] + IG[t])),
   IR'[t] == (\[Beta]*SR[t]*(IG[t] + IR[t])) - ((\[Alpha] + b)*IR[t]),
 
 
   (*call initial conditions*)
   HG[0] == HGinit, HR[0] == HRinit,
   aG[0] == 1.2, qG[0] == (1.2 - b)/KG,
   SG[0] == SGinit, IG[0] == IGinit, RG[0] == RGinit,
   SR[0] == SRinit,
   IR[0] ==
    IRinit                                                            \
        };

sol = NDSolve[eqns, vars, tseries];

{Plot[Evaluate[{SG[t], IG[t], RG[t], SR[t], IR[t]} /. sol], tseries,
  ImageSize -> Large, PlotLegends -> {"SG", "IG", "RG", "SR", "IR"}],
 Plot[Evaluate[{HG[t], HR[t]} /. sol], tseries, ImageSize -> Large,
  PlotLegends -> {"HG", "HR"}]}
<Moderator's note: Please use CODE tags when posting code.>
 
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  • #2
What is your question?
 
  • #3
how do we change aG with time
 
  • #4
You need to make it an actual function of time
Code: 
aG[t] == a0 + a1 * t
with appropriate constants a0 and a1.
 
  • #5
LETS_GO said:
how do we change aG with time
It looks like it already is changing with each time step on line 37:: 
aG[t] == aG[t - 1] - 0.05,(*aG = Greys max. reproductive rate -
   possible birth control??*)
That looks like the change that you want. I'm not familiar with NDsolve and don't see anything wrong with your code.
 
  • #6
FactChecker said:
It looks like it already is changing with each time step on line 37:: 
aG[t] == aG[t - 1] - 0.05,(*aG = Greys max. reproductive rate -
   possible birth control??*)
That looks like the change that you want. I'm not familiar with NDsolve and don't see anything wrong with your code.
That doesn't work, because NDSolve does not iterate the solution like that. It needs to be a proper function of time. (And note that t is a real variable, not an index, so t - 1 means "one unit of time earlier".)
 
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  • #7
DrClaude said:
You need to make it an actual function of time
Code: 
aG[t] == a0 + a1 * t
with appropriate constants a0 and a1.

So I want my starting value for aG to be 1.2 and I want it to go down to at least 0.35 with a decrease of 0.5 for each time step
 
  • #8
LETS_GO said:
So I want my starting value for aG to be 1.2 and I want it to go down to at least 0.35 with a decrease of 0.5 for each time step
But there is no time step! The problem is not fundamentally discrete, but continuous. You need to figure out what aG is as a function of time.
 
  • #9
hmmm I have tried the last couple of days, and im really not sure how to do this
 

FAQ: Trying to model red and grey squirrels

1. What are the key differences between red and grey squirrels that should be considered in a model?

The key differences between red and grey squirrels include their physical characteristics, behavior, habitat preferences, and dietary needs. Red squirrels are generally smaller, have reddish fur, and are more territorial. Grey squirrels are larger, have grey fur, and are more adaptable to different environments. These differences can affect their interactions, competition for resources, and survival rates.

2. How do environmental factors influence the population dynamics of red and grey squirrels?

Environmental factors such as availability of food sources, habitat quality, presence of predators, and climate conditions significantly influence the population dynamics of red and grey squirrels. Grey squirrels tend to thrive in urban and suburban areas with abundant food supplies, whereas red squirrels are more sensitive to habitat changes and tend to prefer coniferous forests. Climate change can also impact their populations by altering the availability of food and suitable habitats.

3. What role do disease and parasites play in the competition between red and grey squirrels?

Disease and parasites play a crucial role in the competition between red and grey squirrels. Grey squirrels are carriers of the squirrelpox virus, which is lethal to red squirrels but does not affect grey squirrels. This disease has significantly contributed to the decline of red squirrel populations in areas where both species coexist. Additionally, parasites such as fleas and ticks can impact the health and survival of both species, but the presence of squirrelpox is a major factor in their competitive dynamics.

4. Can models predict the long-term coexistence or displacement of red squirrels by grey squirrels?

Models can be used to predict the long-term coexistence or displacement of red squirrels by grey squirrels by incorporating various factors such as reproduction rates, mortality rates, disease transmission, and habitat changes. These models often suggest that without intervention, grey squirrels are likely to continue displacing red squirrels due to their competitive advantages and disease resistance. However, conservation efforts and habitat management can influence these outcomes and potentially support the coexistence of both species.

5. What conservation strategies can be derived from modeling the interactions between red and grey squirrels?

Conservation strategies derived from modeling the interactions between red and grey squirrels include habitat management, disease control, and targeted removal of grey squirrels in critical areas. Creating and maintaining habitats that favor red squirrels, such as coniferous forests, can help support their populations. Controlling the spread of squirrelpox through vaccination or other means can reduce the impact of the disease on red squirrels. Additionally, localized removal of grey squirrels in areas where red squirrels are present can help reduce competition and support red squirrel conservation efforts.

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