Modeling Population Dynamics of Flies, Frogs, and Crocodiles

In summary: Consider the case of no frogs, Q(t) = 0 for all t. Will the fate of the crocs then depend in any way on the number of flies (and vice versa)? What does that tell...The system will stay linear because the negative coefficients cancel each other out.
  • #1
Jbreezy
582
0

Homework Statement



Flies, frogs, and crocodiles coexist in an environment. To survive, frogs need to eat flies and crocodiles need to eat frogs. In the absence of frogs, the fly population will grow exponentially and the crocodile population will decay exponentially. In the absence of crocodiles and flies, the grog population will decay exponentially. If P(t) , Q(t), and R(t) represent the populations of these three species at time t, write a system of differential equations as a model for their evolution. If the constants in your equation are all positive, explain why you have used plus or minus signs.

Homework Equations


Not really sure how to do this or where to start at.


The Attempt at a Solution



I said Dp/dt is for my crocs so p is that variable to represent crocidiles.
I said dq/dt is for frogs so q is the variable for frogs and dr/dt is for flies so r is for flies.

I'm having trouble to get going here.
 
Physics news on Phys.org
  • #2
Jbreezy said:

Homework Statement



Flies, frogs, and crocodiles coexist in an environment. To survive, frogs need to eat flies and crocodiles need to eat frogs. In the absence of frogs, the fly population will grow exponentially and the crocodile population will decay exponentially. In the absence of crocodiles and flies, the grog population will decay exponentially. If P(t) , Q(t), and R(t) represent the populations of these three species at time t, write a system of differential equations as a model for their evolution. If the constants in your equation are all positive, explain why you have used plus or minus signs.

Homework Equations


Not really sure how to do this or where to start at.


The Attempt at a Solution



I said Dp/dt is for my crocs so p is that variable to represent crocidiles.
I said dq/dt is for frogs so q is the variable for frogs and dr/dt is for flies so r is for flies.

I'm having trouble to get going here.

You are looking for a linear system with constant coefficients, where
[tex]
\dot P = a_1 P + a_2 Q + a_3 R \\
\dot Q = b_1 P + b_2 Q + b_3 R \\
\dot R = c_1 P + c_2 Q + c_3 R
[/tex]
Now use the given assumptions to determine which coefficients are positive, which are negative, and which are zero.
 
  • #3
pasmith said:
You are looking for a linear system with constant coefficients, where
[tex]
\dot P = a_1 P + a_2 Q + a_3 R \\
\dot Q = b_1 P + b_2 Q + b_3 R \\
\dot R = c_1 P + c_2 Q + c_3 R
[/tex]
Now use the given assumptions to determine which coefficients are positive, which are negative, and which are zero.

But why would the system be linear when in the directions it says that some of them fall die off exponentially or grow exponentially? Just wondering
 
  • #4
Jbreezy said:
But why would the system be linear when in the directions it says that some of them fall die off exponentially or grow exponentially? Just wondering
Suppose we have ##f'(x)=a_1f(x)##. What is ##f##? :wink:
 
  • #5
So like if you integrated it ? I don't get it. Or just f(x) = f(x)' / a1 ??
 
  • #6
Jbreezy said:
So like if you integrated it ? I don't get it. Or just f(x) = f(x)' / a1 ??
Try ##a_1=\frac{f'(x)}{f(x)}##. This is a differential equation you should be able to solve. :tongue:
 
  • #7
This is what I came up with I doubt it is right.

##\dot P = a_1 P + a_2 Q + a_3 R \\
\dot Q = b_1 P + b_2 Q - b_3 R \\
\dot R = -c_1 P - c_2 Q + c_3 R ##
In order of equations;
because crocs decay exponentially if frogs are gone and the flies will grow.
all pos. because if all are working crocs are happy.
Minus b_3 because if there are no crocs or flies the frogs decay
because crocs decay exponentially if frogs are gone and the flies will grow.
 
  • #8
Jbreezy said:
This is what I came up with I doubt it is right.

##\dot P = a_1 P + a_2 Q + a_3 R \\
\dot Q = b_1 P + b_2 Q - b_3 R \\
\dot R = -c_1 P - c_2 Q + c_3 R ##
In order of equations;
because crocs decay exponentially if frogs are gone and the flies will grow.
all pos. because if all are working crocs are happy.
Minus b_3 because if there are no crocs or flies the frogs decay
because crocs decay exponentially if frogs are gone and the flies will grow.
I'm not sure which label represents which species. The OP order is flies, frogs, crocs, so I'll assume that's P, Q, R respectively.
Consider the case of no frogs, Q(t) = 0 for all t. Will the fate of the crocs then depend in any way on the number of flies (and vice versa)? What does that tell you about the coefficients?
 
  • #9
##\dot P = a_1 P + a_2 Q + a_3 R \\
\dot Q = b_1 P + b_2 Q - b_3 R \\
\dot R = -c_1 P - c_2 Q + c_3 R ##



OK your order is correct Haruspex. I redid my equations. I think that the coefficents in those equation should be 0?

##\dot P = a_1 P - a_2 Q \\
\dot Q = b_1 P - b_2 Q + b_3 R \\
\dot R = c_2 Q - c_3 R##
I was just thinking that the flies and crocs are only related through the frogs.
 
  • #10
Think about how crocs affect the frog population. Your equation for ##\dot Q## has the frog population growing as the croc population grows ever larger. Does that make any sense?

For example, suppose a1=a2=P=0, b2=c3=1, and b3=c2=2. There are no flies, yet the population of frogs and crocs grow exponentially.
 
  • #11
D H said:
Think about how crocs affect the frog population. Your equation for ##\dot Q## has the frog population growing as the croc population grows ever larger. Does that make any sense?

For example, suppose a1=a2=P=0, b2=c3=1, and b3=c2=2. There are no flies, yet the population of frogs and crocs grow exponentially.

R is crocs. So in my second equation I have ...

##\dot P = a_1 P - a_2 Q \\
\dot Q = b_1 P - b_2 Q + b_3 R \\
\dot R = c_2 Q - c_3 R##

Frogs are negative in the equation for Q so doesn't that mean they decrease? Oh man I'm confused.
 
  • #12
Why is the relation between increasing frog population (##\dot Q##) and croc population (##b_3 R##) a positive one? Does that make *any* sense? What happens to the poor frogs if there are a lot of crocs around?
 
  • #13
pasmith said:
You are looking for a linear system with constant coefficients
Of course, a linear system will exhibit some unrealistic behaviour. E.g. if R = 0 at some time but Q > 0 then the crocs will come back from extinction.
 
  • #14
haruspex said:
Of course, a linear system will exhibit some unrealistic behaviour. E.g. if R = 0 at some time but Q > 0 then the crocs will come back from extinction.

I would prefer a non-linear system (there really ought to be a non-trivial fixed point corresponding to crocodiles being extinct).

Of course, once a population is sufficiently small the assumption that it can be modeled as a real-valued function of time no longer holds.
 
  • #15
pasmith said:
I would prefer a non-linear system (there really ought to be a non-trivial fixed point corresponding to crocodiles being extinct).

Of course, once a population is sufficiently small the assumption that it can be modeled as a real-valued function of time no longer holds.
Right, but even before that there is the doubtful assumption that the fecundity is directly proportional to the prey/predator ratio and with no time lag.
 
  • #16
haruspex said:
Right, but even before that there is the doubtful assumption that the fecundity is directly proportional to the prey/predator ratio and with no time lag.
Even without taking time lag into account, the frogs don't produce crocs. Crocs have to multiply on their own, and this will need crocs.

Why are we looking for a linear system?
 
  • #17
mfb said:
Even without taking time lag into account, the frogs don't produce crocs. Crocs have to multiply on their own, and this will need crocs.
Yes, that was the point previously discussed. I was noting that even when there are lots of crocs, if there are humungous numbers of frogs then the simple linear relationship will break down.
I agree it is not given that a linear relationship should be assumed, but it does fit with information given about exponential growth and decay.
 
  • #18
The given exponential parts are fine (not completely realistic, but good enough), but that does not say anything about the interaction terms.
 

1. How are the population dynamics of flies, frogs, and crocodiles modeled?

The population dynamics of flies, frogs, and crocodiles are usually modeled using mathematical equations and computer simulations. These models take into account various factors such as birth rate, death rate, migration, and environmental conditions to predict how the populations of these species will change over time.

2. What are some common factors that affect the population dynamics of these species?

The population dynamics of flies, frogs, and crocodiles can be influenced by a variety of factors including food availability, predation, disease, climate change, and human activities such as habitat destruction and pollution. These factors can impact the birth and death rates of these species, as well as their ability to migrate and find suitable habitats.

3. How can modeling population dynamics help us understand and manage these species?

Modeling population dynamics can provide valuable insights into the behavior and trends of these species. By understanding how their populations change over time and what factors contribute to these changes, we can make more informed decisions about conservation and management efforts. These models can also help us predict how these species may respond to different scenarios and interventions.

4. Are there any limitations to modeling population dynamics?

While modeling population dynamics can be a useful tool, it is important to note that these models are simplifications of complex natural systems. They may not always accurately reflect the real-world dynamics of these species and can be affected by uncertainties and assumptions. Therefore, it is important to use multiple models and approaches to gain a more comprehensive understanding of these populations.

5. How can we collect data to improve and validate these population dynamics models?

Collecting accurate and reliable data is crucial for improving and validating population dynamics models. This can be done through field surveys, laboratory experiments, and long-term monitoring of these species. By continuously collecting and analyzing data, we can update and refine these models to better reflect the real-world dynamics of flies, frogs, and crocodiles.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
5K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
3
Replies
73
Views
9K
Back
Top