Physics 212, 2019: Lecture 10 - Revision history

Ilya: /* Your work */

2019-02-17T18:28:25Z

‎Your work

Ilya: /* Nonlinear pendulum */

2019-02-17T18:26:41Z

‎Nonlinear pendulum

Ilya: /* Predator-prey Lotka-Volterra model */

2019-02-17T18:25:40Z

‎Predator-prey Lotka-Volterra model

Ilya: /* Solving systems of ODEs */

2019-02-17T18:23:20Z

‎Solving systems of ODEs

Ilya: /* Your work */

2019-02-17T18:22:30Z

‎Your work

Ilya: /* Solving systems of ODEs */

2019-02-17T04:46:23Z

‎Solving systems of ODEs

Ilya at 04:44, 17 February 2019

2019-02-17T04:44:11Z

Ilya at 04:39, 17 February 2019

2019-02-17T04:39:00Z

Ilya: Created page with "{{PHYS212-2019}} In this lecture, we will introduce models that involved coupled, nonlinear ordinary differential equations, as well as methods for solving them. The first m..."

2019-02-17T04:31:05Z

Created page with "{{PHYS212-2019}} In this lecture, we will introduce models that involved coupled, nonlinear ordinary differential equations, as well as methods for solving them. The first m..."

New page

{{PHYS212-2019}}

In this lecture, we will introduce models that involved coupled, nonlinear ordinary differential equations, as well as methods for solving them.

The first model we will consider is the Malthusian growth with carrying capacity -- where a population grows nearly exponentially fast, before it reaches the maximum population size that can be supported by the environment, <math>dn/dt = r n(n_0-n)</math>, where <math>n_0</math> is known as the carrying capacity. During the previous lecture and the lab, you were supposed to solve the corresponding ODE numerically -- and you will have noticed how the growth starts quickly, and then saturates.

Our second model that we will consider is the predator-prey model of population growth. You can read a lot about such generic systems, including their history, [https://en.wikipedia.org/wiki/Lotka–Volterra_equations here]. Briefly, we consider a prey animal species <math>x</math>, which grows exponentially, but at the same time it's eaten by the predator species, denoted by <math>y</math>. The dynamics of the prey is <math>\frac{dx}{dt}=rx - \alpha xy</math>, where the second term denotes the capture of the prey. Indeed, the more prey animals there are, the easier it is to capture them, and the more predators there are, the more prey they will capture, resulting in the bilinear form <math>\alpha x y</math>. At the same time, when the predators eat, they have enough food to procreate -- but usually killing one prey only produces fewer than one predators. Further, predators also die a natural death. As a result, <math>\frac{dy}{dt}= \beta xy - d y</math>, where <math>\beta<\alpha</math>. This is an interesting dynamical system that exhibits oscillations -- number of preys goes up, they become abundant, this leads to the growth of the number of predators. Predators start eating prey, and collapse their number; then there's an insufficient number of prey to support the predator population, and the latter collapses as well. The whole cycle restarts with the renewed growth of prey at that point.

==Runge-Kutta 2 integration==

===Your work===
;Problem 1: Use a different differential equation from those provided to show that Runge-Kutta 2 has error <math>O(dt^2)</math>, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it. Do this in a Jupyter notebook.

===Scripts for this Lecture===
::[[media:growthfunctions2019.txt | GrowthFunctions]] -- module defining different growth functions.
::[[media:Integrators2019.txt | Integrators]] -- module defining different integrators (Euler and Runge-Kutta, which we will use during the next lecture).
::[[media:moduleTwo2019.txt | Module 2]] -- catch-all scripts I used during the class.
::[[media:IntegrateODE.txt | Module 2 as a notebook]] -- integration as a notebook.

@@ Line 28: / Line 28: @@
 ;Problem 2: Use the same extension of the RK2 algorithm to solve the nonlinear pendulum problem. Compare the results to the built-in ODE solver.
-===Scripts for this Lecture===
+==Scripts for this Lecture==
 ::[[media:SystemsOfODEs.txt|Solving Systems of ODEs]] Jupyter notebook

@@ Line 13: / Line 13: @@
 ==Nonlinear pendulum==
 Consider a non-idealized pendulum, which undergoes oscillations with a non-infinitesimal amplitude. Then the restoring torque that it experiences is <math>M=mgl \sim \theta</math>, where <math>m</math> is the mass, <math>l</math> is the pendulum length, and <math>\theta</math> is the angle its axis forms with the vertical. Newton's equations of motion give for this system <math>\frac{d^2\theta}{dt^2}=gl\sin\theta</math>. This seems like a single equation, but it is second order in the temporal derivative. It is then equivalent to two coupled first order differential equations <math>\omega =\frac{d\theta}{dt}</math> and <math>\frac{d\omega}{dt^2}=gl\sin\theta</math>, where <math>\omega</math> is the angular velocity
 ==Solving systems of ODEs==

@@ Line 8: / Line 8: @@
 ==Predator-prey Lotka-Volterra model==
 Our second model that we will consider is the predator-prey model of population growth. You can read a lot about such generic systems, including their history, [https://en.wikipedia.org/wiki/Lotka–Volterra_equations here]. Briefly, we consider a prey animal species <math>x</math>, which grows exponentially, but at the same time it's eaten by the predator species, denoted by <math>y</math>. The dynamics of the prey is <math>\frac{dx}{dt}=rx - \alpha xy</math>, where the second term denotes the capture of the prey. Indeed, the more prey animals there are, the easier it is to capture them, and the more predators there are, the more prey they will capture, resulting in the bilinear form <math>\alpha x y</math>.  At the same time, when the predators eat, they have enough food to procreate -- but usually killing one prey only produces fewer than one predators. Further, predators also die a natural death. As a result, <math>\frac{dy}{dt}= \beta xy - d y</math>, where <math>\beta<\alpha</math>. This is an interesting dynamical system that exhibits oscillations -- number of preys goes up, they become abundant, this leads to the growth of the number of predators. Predators start eating prey, and collapse their number; then there's an insufficient number of prey to support the predator population, and the latter collapses as well. The whole cycle restarts with the renewed growth of prey at that point. One can think of more complicated similar systems of equations, but we will leave this to the homework project.
 ==Nonlinear pendulum==

@@ Line 17: / Line 17: @@
 We will observe the oscillations, and how they can have different amplitudes depending on the initial condition, but always are around the same point.
-===Your work===
+==Additional Python concepts==
 ;Problem 1: Write an extension of the RK2 algorithm to a system of coupled ODEs of an arbitrary size and use it to solve the Lotka-Volterra model. Compare the results to the built-in ODE solver. Start with the multidimensional version of the Euler algorithm Jupyter notebook.
 ;Problem 2: Use the same extension of the RK2 algorithm to solve the nonlinear pendulum problem. Compare the results to the built-in ODE solver.

@@ Line 18: / Line 18: @@
 ===Your work===
-;Problem 1: Write an extension of the RK2 algorithm to a system of coupled ODEs of an arbitrary size ands it to solve the Lotka-Volterra model. Compare the results to the built-in ODE solver.
+;Problem 1: Write an extension of the RK2 algorithm to a system of coupled ODEs of an arbitrary size and use it to solve the Lotka-Volterra model. Compare the results to the built-in ODE solver. Start with the multidimensional version of the Euler algorithm Jupyter notebook.
 ;Problem 2: Use the same extension of the RK2 algorithm to solve the nonlinear pendulum problem. Compare the results to the built-in ODE solver.
 ===Scripts for this Lecture===
 ::[[media:SystemsOfODEs.txt|Solving Systems of ODEs]] Jupyter notebook

@@ Line 1: / Line 1: @@
 {{PHYS212-2019}}
-In this lecture, we will introduce models that involved coupled, nonlinear ordinary differential equations, as well as methods for solving them.
+To date, we only considered simple dynamics, which has a single dynamical variable, whose first temporal derivative is determined by the variable itself. In this lecture, we will introduce models that involved coupled, nonlinear ordinary differential equations, as well as methods for solving them.
 The first model we will consider is the Malthusian growth with carrying capacity -- where a population grows nearly exponentially fast, before it reaches the maximum population size that can be supported by the environment, <math>dn/dt = r n(n_0-n)</math>, where <math>n_0</math> is known as the carrying capacity. During the previous lecture and the lab, you were supposed to solve the corresponding ODE numerically -- and you will have noticed how the growth starts quickly, and then saturates.
-Our second model that we will consider is the predator-prey model of population growth. You can read a lot about such generic systems, including their history, [https://en.wikipedia.org/wiki/Lotka–Volterra_equations here]. Briefly, we consider a prey animal species <math>x</math>, which grows exponentially, but at the same time it's eaten by the predator species, denoted by <math>y</math>. The dynamics of the prey is <math>\frac{dx}{dt}=rx - \alpha xy</math>, where the second term denotes the capture of the prey. Indeed, the more prey animals there are, the easier it is to capture them, and the more predators there are, the more prey they will capture, resulting in the bilinear form <math>\alpha x y</math>.  At the same time, when the predators eat, they have enough food to procreate -- but usually killing one prey only produces fewer than one predators. Further, predators also die a natural death. As a result, <math>\frac{dy}{dt}= \beta xy - d y</math>, where <math>\beta<\alpha</math>. This is an interesting dynamical system that exhibits oscillations -- number of preys goes up, they become abundant, this leads to the growth of the number of predators. Predators start eating prey, and collapse their number; then there's an insufficient number of prey to support the predator population, and the latter collapses as well. The whole cycle restarts with the renewed growth of prey at that point.
+==Predator-prey Lotka-Volterra model==
+Our second model that we will consider is the predator-prey model of population growth. You can read a lot about such generic systems, including their history, [https://en.wikipedia.org/wiki/Lotka–Volterra_equations here]. Briefly, we consider a prey animal species <math>x</math>, which grows exponentially, but at the same time it's eaten by the predator species, denoted by <math>y</math>. The dynamics of the prey is <math>\frac{dx}{dt}=rx - \alpha xy</math>, where the second term denotes the capture of the prey. Indeed, the more prey animals there are, the easier it is to capture them, and the more predators there are, the more prey they will capture, resulting in the bilinear form <math>\alpha x y</math>.  At the same time, when the predators eat, they have enough food to procreate -- but usually killing one prey only produces fewer than one predators. Further, predators also die a natural death. As a result, <math>\frac{dy}{dt}= \beta xy - d y</math>, where <math>\beta<\alpha</math>. This is an interesting dynamical system that exhibits oscillations -- number of preys goes up, they become abundant, this leads to the growth of the number of predators. Predators start eating prey, and collapse their number; then there's an insufficient number of prey to support the predator population, and the latter collapses as well. The whole cycle restarts with the renewed growth of prey at that point. One can think of more complicated similar systems of equations, but we will leave this to the homework project.
 ===Your work===