33
votes
Accepted
Turing patterns
I developed a reaction-diffusion-advection model of pattern formation in semi-arid vegetation (tiger bush) 20 years ago, which shows a type of Turing instability. Plants ($n$) consume water ($w$) and ...
- 20.4k
24
votes
Simulating a partial differential equation - reaction-diffusion systems and Turing patterns
You're really close. Just a couple things: 1) your initial conditions are spatially uniform random numbers, which will prevent pattern formation, and 2) according to the linked post, the domain ...
- 20.4k
22
votes
Turing patterns
I did some work with the Brusselator some time ago. This is the reaction-diffusion equations which generate Turing patterns. There are some things you need to know:
(1) The non-linear PDEs have ...
- 2,914
19
votes
Accepted
Nonlinear dispersal equation modeling insect aggregation
If the equation is correct, then it's probably another example that we need special treatment for the discretization of conservation law.
As mentioned in the comment above, one easy-to-notice issue of ...
- 68.4k
18
votes
The Lotka-Volterra predator-prey model
The dimensional phase-space plot
The analysis of the predator-prey model is incomplete without the phase-space plot. So, here it is,
...
- 12k
15
votes
Accepted
Rosenzweig-MacArthur predator-prey model
Note
In my attempt to answer the OP's question, I have presented all most all the visuals/graphs which are important for the analysis of such models. If there is something missing or physically ...
- 12k
13
votes
Solving the Lotka-McKendrick model with NDSolve
I'm not an expert on age-structured populations (particularly this continuous-time model) and I know better numerical methods exist, but why not just discretize in age ...
- 20.4k
10
votes
Mathematica code for bifurcation diagram in 3D
Looks like a little three-species food web model -- a perfect excuse to use my new EcoEvo package.
First, install the package (one-time only):
...
- 20.4k
9
votes
Sensitivity analysis of parameter on eigenvalues of predator-prey model
Here's a solution using my EcoEvo package, which is designed for just this kind of problem. First, install the package (only need to do this once):
...
- 20.4k
9
votes
Accepted
Steady state solution (1D) of nonlinear dispersal equation
The problem can be solved analytically.
First we transform the equation a bit. Integrate the ODE once we obtain
...
- 68.4k
7
votes
System of nonlinear PDE 2D (Reaction-Diffusion type) with periodic boundary condition
As @xzczd alluded to, your code is almost correct. The message NDSolve::mxsst is more of a note than an error or even a warning.
Here's a working version because ...
- 20.4k
7
votes
Solving the Lotka-McKendrick model with NDSolve
There is no unique solution for data provided by @Pillsy, since boundary and initial conditions are inconsistent. To show it we just use exact solution in a form:
...
- 48.8k
7
votes
How can I speed up NDSolve
There are no gradients in x and y at L> 10. Nothing happens when ...
- 48.8k
7
votes
Accepted
How to get the steady-state analytical solution of a set of nonlinear differential equations?
A steady-state is a condition in which the concentrations do not change anymore. This means that $$\frac{\mathrm{d}[X]}{\mathrm{d}t} = 0$$ for every component $X$.
Let's first change your equations to ...
- 33.4k
6
votes
Bifurcation diagram for ODEs
Here's an analysis using a new version of my EcoEvo package.
First, install the latest version of the package (needed only once):
...
- 20.4k
6
votes
Accepted
1-D prey-predator system with diffusion and time-dependent parameters in Mathematica 12
After minor corrections and typo removing's we have stable result with some options:
...
- 48.8k
6
votes
Accepted
6
votes
Simple Birth Death Process
I just did this in class today using @IstvánZachar's GillespieSSA function from this answer. Load that, then:
...
- 20.4k
6
votes
5
votes
Accepted
Simple Birth Death Process
How can I store Δt = Min[Δt1, Δt2] so that I can plot time vs population
One way is
...
- 401k
5
votes
Accepted
Trying to recreate a plot which tracks the age of individuals inside a compartment model
The average time spent within a compartment is the reciprocal of the per capita rate an individual leaves the compartment. In this case, the average time spent in H...
- 20.4k
5
votes
Accepted
Numerically solve delay ODE
I'm not sure the separate NDSolve to initialize the initial conditions is even necessary. If you just use your second NDSolve ...
- 20.4k
5
votes
Nonlinear dispersal equation modeling insect aggregation
If only the steady state is desired, it can be obtained easily by
sa = Values[DSolve[1 + u[x] D[u[x], x] == 0, u[x], x] /. C[1] -> c][[2, 1]]
and ...
- 62.1k
4
votes
Solving the Lotka-McKendrick model with NDSolve
To give some convincing publication to the round of answerers and the owner of the question: Numerical methods for the Lotka–McKendrick’s equation
Galena Pelovska, Mimmo Iannelli∗
Dipartimento di ...
- 4,126
4
votes
How to get the steady-state analytical solution of a set of nonlinear differential equations?
The steady state solution is a solution, where all functions become constant.
That is, all y' are to be set to zero
While the standard routines like Solve, Reduce, Eliminate are going to run endlessly,...
- 4,628
3
votes
How solve a PDE system with Specific Initial Condition?
I don't have time right now to fully solve the problem, but there is an issue with the way you are describing the conditionals. You need to flip some signs and use some ...
- 1,858
3
votes
How to perform integration processes NDSOlve and show list of random variables used in this process?
Your attempt in (i) fails because you forgot to initialize ics for every round of repeation.
Generating the list of ξ[j, k] ...
- 68.4k
3
votes
How to solve an ODE system that periodically increases in size
This looks like a Lotka-Volterra competition model with weak, symmetric interspecific competition. If I understand correctly, you want to build up a community by introducing new species every ...
- 20.4k
3
votes
Harvested prey–predator model incorporating a prey refuge
Some modifications for the code in Vitaliy Kaurov's answer:
<<MaTeX`
<< c:\CurvesGraphics6\CurvesGraphics6.m
Initial condition outside the limit cycle:
...
- 31.3k
3
votes
Accepted
Poincaré section for non-autonomous logistic equation with periodic harvesting
Question 1: Read the manual about "Reap". Here is the working code:
...
- 56.4k
Only top scored, non community-wiki answers of a minimum length are eligible