Generalized Lotka–volterra Equation Puns

How do I write the system of equations when there are 4 species:

H: Humans which eat shark and fish

S: Shark which eat humans and dolphins

D: Dolphins which eat fish

F: Fish which eat plankton.

Here, plankton is assumed to be an infinite resource.

I get the logic of writing the equations for 2 species which are predator and prey, but how would I show the relationship between sharks eating humans/humans eating shark and the population of fish increasing/decreasing??

I've written:

dF/dt = F(a-bD-cH)

dH/dt = H(-k+(d-e)S+fF)

dS/dt = S(-g+hD+(j-p)H)

dD/dt = D(m-nS+oF)

where a,b,c,d,e,f,g,h,j,k,m,n,o,p are positive constants.

does this make sense? I couldn't find any info on this non-linear competition format. Should I take fish as "competing" against humans, or sharks? Any links on this would be helpful

I'm just starting to learn this concept so sorry if it does not make any sense, thanks in advance.

Hello. I am currently doing an investigation on the Lotka-Volterra model for predator prey dynamics. I am trying to fit the model (a pair of 2-dimensional non-linear differential equations) to real time series data of lynx and hare populations. (Link to both of these)

https://imgur.com/a/FfaJaB7

To do this I need to find values for a,b,c,d which best match the dynamics of the populations, so I want to try imperically determine this from the data. Does anyone know a good method for doing this? I tried substituting values of x and y (population numbers) over multiple years then setting dx/dt and dy/dt to the change in populations between those years then just solving the system of equations but I got values which were just flat out impossible such as the growth of the populations being exponentially negative, and this can't be pinned on unreliable data, its just that my method of calculating it was wrong.

If anyone could suggest a method for determining the coefficients a,b,c and d from the data I would greatly appreciate it. I hope I have explained this well enough, let me know if anything should be clarified, thanks!

We had this Lotka Volterra system that we were analyzing in Dynamical Systems, and our professor used it as sort of a way to introduce Hamiltonian systems to our exercise, to show that in Hamiltonian systems, the only critical points are saddles and centers. I noticed that the change in variables we used is continuous and invertible, and that seemed similar to a definition we saw earlier about topological equivalence or conjugacy. Is this the case in this situation?

Is there any way to apply related rates to the Lotka-Volterra equations in order to predict the changes in the populations at some time?

Hello redditors of /r/askscience,

I'm currently struggling with the Lotka-Volterra equation. I have to create a simulation of it with the program Consideo and therefore have to understand the equation. But the explanation isn't that simple to understand and other explanations, for example by local colleges aren't that simple either :(

So I thought that you guys and girls of /r/askscience may be smart enough to explain the Lotka-Volterra equation to me.

Thanks in advance,

/u/s22l1g11.

Got a worksheet today that is sadly due tomorrow and having only had one lecture on differential equations and having an exam tomorrow I don't feel I have time to figure this out from the textbook alone. Any guidance on how to proceed would be very much appreciated. We're given: dR/dt=αR-bRW dW/dt=cRW-gW a=0.5, b=0.8, c=0.6, g=0.75 We're supposed to figure out:

1. what are R&W when the population is constant -I know that population being constant means dR/dt=dWdt=0, but I'm stuck from there 2)use the chain rule to derive a formula for dR/dW

While trying to learn Lotka-Volterra Model by myself I encountered a point I didn't understand. Here is a link (https://i.imgur.com/MFZOmrB.jpg) to picture from Hofbauer's and Sigmund's Evolutionary Games and Population Dynamics, that I have been studying. After the fixed point/rest point F = (x,y) = (c/d, a/b) (x and y with overline in the linked picture) is found, the authors give proof that the solutions to the differential equation system x'(t) = x(t)(a - by); y'(t) = y(t)(-c + dx) (system is called (2.1) in the picture) are periodic/cyclic.

I understand everything up until the authors introduce the new function V(x,y). I think I understand where it comes from, but I don't understand how it is connected with the starting differential equations (2.1), or why it tells us the information that the solutions are periodic when the V(x,y) equals to some constant C. Additionally, I'm not sure what the maxima of functions H(x) and G(y) has to do with anything.

If someone could explain what happens after (2.12) in the picture, point me to a better source or a place where someone can answer my question, I would be grateful.

Edit: I think I got it. The fundamental problem was that I didn't know what contour line meant. I understand that contour line tells us how the function V behaves with different real numbers C. By changing the Lotka-Volterra differential equation system (2.1) to single line equation V, we can find its contour lines and determine what it tells us about the system (2.1).

Hi, I’m having trouble progressing with my Math IA (basically math essay) and need access to some sort of site or way to get data on the populations of two species. My project is basically evaluating different population models (Lotka Volterra, logistic model, competitive Lotka Volterra etc...) and comparing to real results - which is what I need a database for.

I need the population of a predator and prey species over time (preferably many generations), obviously ones that interact with each other, but 10 hours of searching has yielded me no results. I genuinely cannot find anything. I would really appreciate if anyone could share resources for this kind of thing.

For example, a dataset on population of a Canadian Lynx and Snowshoe Hare would be perfect, but really I don’t care what species they are, I can adapt my paper.

As a side question (not sure if I’m allowed to ask those) How is carrying capacity in an ecosystem determined? How could I find this capacity on a predator prey relationship? (See Competitive Lotka-Volterra Equations.)

What about the effect of one species on another, how is that determined? In the competitive equation it is specified as “a”, so I presume it is just a variable but am unsure on how that variable is gotten.

Thank you everyone!

Predator-prey reversal: "In marine benthic communities, rock lobsters, Jasus lalandii, usually represent a keystone predator in diverse ecosystems. However, if its common prey, the whelk (Burnupena sp.), greatly outnumbers the rock lobster, it may result in an alternative stable ecosystem where rock lobsters are killed by whelks, which dominate in the alternated ecosystem."

What if this system were bistable and looped indefinitely?

The basic schematics of this system remind me of the alternation between left and right political dominance over time due to changes in a political landscape which shifts between two stable political paradigms, with the intermediate stance being centrism.

The point at which the left-right political paradigm reversal is initiated might be explained by some kind of population threshold which, when reached, causes the 'predator' entity to become complacent and decline, or the 'prey' entity to adapt to the new paradigm and compete until reversal is achieved.

If this kind of model can represent the left right political system, then there are three possible outcome cases (I think):

1. After some period of alternation between states, one of the populations consumes the other entirely resulting in a single population in the system (A dominant-party political system which will devolve into a de jure, or defacto, single-party system). [Left XOR Right]

2. Both populations survive, alternating between the two states indefinitely (A two-party political system much like USA, and somewhat Australia today. The two major parties require eachother for survival, yet still compete against eachother. This occurs when the left attacks the right and the right attacks the left in a bistable system). [Left OR Right]

3. Both populations decline while alternating over time until they both approach zero (A two-party political system in which both parties decline in popularity until both parties are eventually phased out. In a political system, this phase-out would be achieved with either mutualism between the left and right allowing centrism to establish dominance, or both are phased out to make room for some external party through colonisation, mass migration or conquest). [Left XNOR Right]

Any ideas on what might trigger the reversal of left and right political roles? My guess is that counter-culture plays a HUGE role in this. People get really sick of repetition of trends, and eventually need a change in paradigm (to regulate between diversity

Hello everyone!
In looking for ideas for a Math HL IA, I found the Lotka volterra equation, that models the relationship between predators and prey in an ecosystem.

My teacher said it would be a feasible topic, but I don't really know how I could proceed to explore it.

Any suggestion to get started with it?

Rolling down my list of alliances from EVE's past, this time I'll be asking about what I understood was one of EVE's big powerhouses in the early days of Sov.

As far as I know, they were an ally of BOB and were part of ASCNs destruction, though I haven't heard much and I very well could be wrong. Anyone from those days around to explain their history and fate in a way the wiki can't?

And did I even spell that right?

Hi,

This is homework. Here is my code:

function predprey2

% predprey: MATLAB function that takes an initial guess of the parameter

% values for the predator prey equation and returns the best fitting

% parameter values based on the provided data

t0 = 0; % initial time

tfinal = 15; % final time

Summer0 = 30; % initial condition

Winter0 = 20; % initial condition

guess = [2.01; 2; 1.98; 2.2]; % guess initial parameter values

actual_params = [2; 2; 2; 2]; % the parameters to build the synthetic data with

% create data for Summer and Winter

[years,y] = ode45(@pfunc,[t0 tfinal], [Summer0, Winter0], [], actual_params);

% TODO add noise to data

ht = y(:, 1);

lt = y(:, 2);

Summer = ht; % experiemntal values prey

Winter = lt; % exoerimental values predator

% calculate best fit paramemters

options = optimoptions(@lsqnonlin, 'Algorithm', 'levenberg-marquardt', 'Display', 'iter');

% anon function parameterize over p on experimental data to find best fit p

fun = @(p)pfunc(years, [Summer; Winter], p);

p = lsqnonlin(fun, guess, [], [], options);

end

function value = pfunc(t, y, p)

% lotka-volterra model

value=[p(1)*y(1)-p(2)*y(1)*y(2);-p(3)*y(2)+p(4)*y(1)*y(2)];

end

The first few lines are for user input. Since I do not have real data to test the model against, I generate data first using the parameter values specified in actual_params and place it in variables Summer, Winter. I use lsqnonlin and use the lotka volterra system of equation as my parameter function. I feel that the way I am doing this is incorrect because the model is specifying derivatives, y' values, and I am giving it y values. However, on the site, it says to not use an error function like least squares because that is already built in. I consider writing an anonymous function, model_y, which I have commented out, to solve the system, but lsqnonlin says that it only accepts functions that return doubles.

I just need a simple, basic explanation of this theory. A google search has yielded nothing but scholarly articles full of jargon.

A couple of friends and me wanted to create a mathematical model of a population of algae and paramecia. We wanted to determine the constants practically using actual algae and paramecia. For practical reasons we were wondering how long the cycles take, as to know how much time we will need for the experiments. To practice with constructing the model we were wondering whether there is any place online where you can find known Lotka-Volterra constants.

Edit: there was one final question that I forgot to put in the title: do you know any cool stuff that we could do with our model after we make it? We already found that we can plot predator and prey density against one another and get circles, and we found here: http://mathnathan.com/2010/12/lv-visual/ that you can also create a map of where the model would go with all values for predator and prey density. I'm looking forward to your ideas!

Hello. I am currently doing an investigation on the Lotka-Volterra model for predator prey dynamics. I am trying to fit the model (a pair of 2-dimensional non-linear differential equations) to real time series data of lynx and hare populations. (Link to both of these)

https://imgur.com/a/FfaJaB7

To do this I need to find values for a,b,c,d which best match the dynamics of the populations, so I want to try imperically determine this from the data. Does anyone know a good method for doing this? I tried substituting values of x and y (population numbers) over multiple years then setting dx/dt and dy/dt to the change in populations between those years then just solving the system of equations but I got values which were just flat out impossible such as the growth of the populations being exponentially negative, and this can't be pinned on unreliable data, its just that my method of calculating it was wrong.

If anyone could suggest a method for determining the coefficients a,b,c and d from the data I would greatly appreciate it. I hope I have explained this well enough, let me know if anything should be clarified, thanks!

Predator-prey reversal: "In marine benthic communities, rock lobsters, Jasus lalandii, usually represent a keystone predator in diverse ecosystems. However, if its common prey, the whelk (Burnupena sp.), greatly outnumbers the rock lobster, it may result in an alternative stable ecosystem where rock lobsters are killed by whelks, which dominate in the alternated ecosystem."

What if this system were bistable and looped indefinitely?

The basic schematics of this system remind me of the alternation between left and right political dominance over time due to changes in a political landscape which shifts between two stable political paradigms, with the intermediate stance being centrism.

The point at which the left-right political paradigm reversal is initiated might be explained by some kind of population threshold which, when reached, causes the 'predator' entity to become complacent and decline, or the 'prey' entity to adapt to the new paradigm and compete until reversal is achieved.

If this kind of model can represent the left right political system, then there are three possible outcome cases (I think):

1. After some period of alternation between states, one of the populations consumes the other entirely resulting in a single population in the system (A dominant-party political system which will devolve into a de jure, or defacto, single-party system). [Left XOR Right]

2. Both populations survive, alternating between the two states indefinitely (A two-party political system much like USA, and somewhat Australia today. The two major parties require eachother for survival, yet still compete against eachother. This occurs when the left attacks the right and the right attacks the left in a bistable system). [Left OR Right]

3. Both populations decline while alternating over time until they both approach zero (A two-party political system in which both parties decline in popularity until both parties are eventually phased out. In a political system, this phase-out would be achieved with either mutualism between the left and right allowing centrism to establish dominance, or both are phased out to make room for some external party through colonisation, mass migration or conquest). [Left XNOR Right]

Any ideas on what might trigger the reversal of left and right political roles? My guess is that counter-culture plays a HUGE role in this. People get really sick of repetition of trends, and eventually need a change in paradigm (to regulate between diversity

I'm not very good with terminology, but does anyone know if there is a lotka-volterra model in which the prey switches positions in the heirarchy with the predator at some point, and it keeps alternating like this between two states.

I'm mostly interested in two cases.

1. after some period of alternation between states, one of the populations consumes the other entirely resulting in a single population in the system.

2. both populations survive, alternating between the two states indefinitely.

Bonus points if you can link me to an example of a model like this, graph or raw data, i don't mind.

edit:

I almost have it. Almost. Symmetrical Intraguild Predation. But this doesn't account for functional change from predator to prey or vice versa based on some threshold. I wonder if bistable systems where the predator and prey keep switching even exists in nature.

I'm still interested if anyone has seen any nice data of systems like this.

https://www.nature.com/articles/s41598-017-07339-w/

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4448886/

"In marine benthic communities, rock lobsters, Jasus lalandii, usually represent a keystone predator in diverse ecosystems. However, if its common prey, the whelk (Burnupena sp.), greatly outnumbers the rock lobster, it may result in an alternative stable ecosystem where rock lobsters are killed by whelks, which dominate in the alternated ecosystem." ... Just like the alternation between left and right political dominance over time due to changes in a political landscape which shifts between two stable political paradigms, with the intermediate stance being centrism.