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Paras31

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1 years, 91 days
Hellenic Open University
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Teacher of Mathematics with a proven track record of working in education management. Proficient in Ease of Adaptation, Course Design, and Instructional Technology. Holds a Bachelor's degree in Mathematics from the University of the Aegean and a Master's in Applied Mathematics at the Hellenic Open University, focusing on Ordinary and Partial Differential Equations. His enthusiasm lies in the application of mathematical models to real-world contexts, such as epidemiology and population growth. Aside from his passion for teaching, Athanasios enjoys football, basketball, and spending time with his dogs.

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These are questions asked by Paras31

SI Model Simulation: Error in Jacobian M...

Paras31 245 Maple 2024
matrix syntax linearalgebra
May 02 2024
1 0

SI_MODEL.mw
I have tried the following code to simulate the following SI Model. But when I run it, it gives me several mistakes and most importantly does not calculate the Jacobian Matrix and the eigenvalues. 

Any help would be deeply appreciated


Many of the most interesting dynamics in nature have to do with interactions between organisms. These interactions are often subtle, indirect, and difficult to detect. Interactions in which one organism consumes all or part of another. This includes predator-prey, herbivore-plant, and parasite-host interactions. These linkages are the prime movers of energy through food chains. They are an important factor in the ecology of populations, determining the mortality of prey and the birth of new predators.
​
Mathematical models and logic suggest that a coupled system of predator and prey should cycle: predators increase when prey is abundant, prey are driven to low numbers by predation, the predators decline, and the prey recover, ad infinitum. One such model that simulates predator-prey interactions is the Lotka - Volterra Model.

 


We will discuss a behavior of 2 D - subsystems: SI - model.


SI Model (without predator)

 

We will study a mathematical model that appears in eco - eco-epidemiology.


The SI model describes the interactions between S-prey with density x and I-prey with density y under the assumption z≡0


Break the population into compartments.

• 

Prey (S)
• 

Infected prey (I)


SI Equations :

(D(x))(t) = r*x(t)-b*x(t)^2-c*x(t)*y(t)-beta*x(t)*y[t]/(a+x(t))

(D(y))(t) = -mu*y(t)+beta*x(t)*y(t)/(a+x(t))

– 

 r stands for the intrinsic growth rate of S-prey.
– 

b defines the intra-class competition in S-prey.
– 

c characterizes the inter-class competition between S-prey and I-prey.
– 

μ stands for the mortality of the I-prey
– 

β considered as the bifurcation parameter.
• 

Note: The dynamics of the system depend on parameter β

 

Modeling with Differential Equations

 

• 

We will fix the parameters in the study to have the following values r = 1, b = 1, c = 1/100, μ = 4/10, a=1/2.75.
• 

The initial values will be x(0)=0.2, y(0)=0.05


As mentioned above the dynamics of the system depend on parameter β. So we will consider two cases when β=0.75 and β=1


We will solve the system for the above values of parameters and initial conditions and for a time interval [0,200] , then we will create its plots

 

Case 1

  restart

beta := .75; r := 1; b := 1; c := 1/100; mu := 4*(1/10); a := 1/2.75; f := r*x(t)-b*x(t)^2-c*x(t)*y(t)-(3/4)*x(t)*y(t)/(a+x(t)); g := -mu*y(t)+(3/4)*x(t)*y(t)/(a+x(t)); deq1 := diff(x(t), t) = f; deq2 := diff(y(t), t) = g; equilibrio := solve({f = 0, g = 0}, {x(t), y(t)}); jacobian := simplify(Matrix(2, 2, [[diff(f, x(t)), diff(f, y(t))], [diff(g, x(t)), diff(g, y(t))]])); A := simplify(eval(jacobian, equilibrio[1])); B := simplify(eval(jacobian, equilibrio[2])); Q := simplify(eval(jacobian, equilibrio[3])); eigen1 := Eigenvalues(A); eigen2 := Eigenvalues(B); eigen3 := Eigenvalues(Q); soln := dsolve({deq1, deq2, x(0) = .2, y(0) = 0.5e-1}, numeric, output = listprocedure); plots:-display(plots:-odeplot(soln, [t, x(t)], 0 .. 200, color = blue, legend = ["x(t)"]), plots:-odeplot(soln, [t, y(t)], 0 .. 200, color = red, legend = ["y(t)"]), labels = ["t", "Population"], title = "Population Dynamics")

  

NULL

 

 

Case 2

restart``

beta := 1; r := 1; b := 1; c := 1/100; Mu := 4*(1/10); a := 1/2.75; f := r*x(t)-b*x(t)^2-c*x(t)*y(t)-x(t)*y(t)/(a+x(t)); g := -Mu*y(t)+x(t)*y(t)/(a+x(t)); deq1 := diff(x(t), t) = f; deq2 := diff(y(t), t) = g; equilibrio := solve({f = 0, g = 0}, {x(t), y(t)}); jacobian := Matrix(2, 2, [[diff(f, x(t)), diff(f, y(t))], [diff(g, x(t)), diff(g, y(t))]], simplify); J_at_equilibrio := [eval(jacobian, equilibrio)]; eigen1 := Eigenvalues(A); eigen2 := Eigenvalues(B); eigen3 := Eigenvalues(Q); soln := dsolve({deq1, deq2, x(0) = .2, y(0) = 0.5e-1}, numeric); plots:-display(plots:-odeplot(soln, [t, x(t)], 0 .. 200, color = blue, legend = ["x(t)"]), plots:-odeplot(soln, [t, y(t)], 0 .. 200, color = red, legend = ["y(t)"]), labels = ["t", "Population"], title = "Population Dynamics with Beta = 1")

  

NULL


 

Download SI_MODEL.mw

 

 

Maple Packages as a Graduate Student...

Paras31 245 Maple
version student
April 29 2024
1 2

Good evening!

I am Athanasios Paraskevopoulos, a graduate student specializing in applied mathematics. Recently, I've started exploring Maple through its trial version and I'm considering making a purchase. My question for you all is: Am I restricted to buying only the graduate student package, or am I free to choose from any of the Maple packages available? Any guidance or personal experiences with different packages would be greatly appreciated!

Thank you in advance for your help!

Plotting Linear System of Equations...

Paras31 245 Maple 2024
linear linear-algebra plotting
April 27 2024
3 0

Hello, everyone,

I am new to Maple and I am trying to get use of it.

I tried to plot the following linear systems in different ways. I realized that the Student Linear Algebra is not as flexible as Linear Algebra. My question is the following. Is there any other way to create a plot without defining the implicit plots?

> 

with(Student[LinearAlgebra])
> 

A := Matrix([[1, 1], [12, 16]]); b := Vector([10, 136]); sol := LinearSolve(A, b)

Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 12, (2, 2) = 16})

  

Vector(2, {(1) = 10, (2) = 136})

  

Vector[column](%id = 36893488153382714652)

 (1)
> 

LinearSystemPlot({x+y = 10, 12*x+16*y = 136}, axes = normal)

 
> 

restart

 

 
> 

with(Student[LinearAlgebra])
> 

A := Matrix([[2, -1, 1], [0, 1, 3], [0, 0, 1]]); b := Vector([-5, 7, 2]); sol := LinearSolve(A, b)

Matrix(3, 3, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 1, (2, 3) = 3, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})

  

Vector(3, {(1) = -5, (2) = 7, (3) = 2})

  

LinearSolve(Matrix(%id = 36893488151878044716), Vector[column](%id = 36893488151878030628))

 (2)
> 

LinearSystemPlot({w = 2, y+3*w = 7, 2*x-y+w = -5}, axes = normal)

 
> 

restart
> 

with(plots); with(LinearAlgebra); A := Matrix([[1, 1], [12, 16]]); b := Vector([10, 136]); sol := LinearSolve(A, b); eq1 := x+y = 10; eq2 := 12*x+16*y = 136; plot1 := implicitplot(eq1, x = -5 .. 10, y = -50 .. 50, color = "red", thickness = 2, labels = ["x", "y"]); plot2 := implicitplot(eq2, x = -5 .. 10, y = -50 .. 50, color = "blue", thickness = 2); display(plot1, plot2, title = "Plot of Linear System", legend = [x+y = 10, 12*x+16*y = 136])

Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 12, (2, 2) = 16})

  

Vector(2, {(1) = 10, (2) = 136})

  

Vector[column](%id = 36893488153330030820)

  

x+y = 10

  

12*x+16*y = 136

  

 
> 

A := Matrix([[2, -1, 1], [0, 1, 3], [0, 0, 1]]); b := Vector([-5, 7, 2]); solution := LinearSolve(A, b); eq1 := 2*x-y+z = -5; eq2 := y+3*z = 7; eq3 := z = 2; plot1 := implicitplot3d(eq1, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, color = "red", style = surface); plot2 := implicitplot3d(eq2, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, color = "blue", style = surface); plot3 := implicitplot3d(eq3, x = -10 .. 10, y = -10 .. 10, z = -10 .. 10, color = "green", style = surface); display(plot1, plot2, plot3, title = "3D Plot of Linear System", axes = boxed)

Matrix(3, 3, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 1, (2, 3) = 3, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})

  

Vector(3, {(1) = -5, (2) = 7, (3) = 2})

  

Vector[column](%id = 36893488152610632156)

  

 
> 

NULL


 

Download linear_systems.mw

 

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