## Profiles on Profiles on Profiles

This week started off with brainstorming questions to see where we wanted to go with the solution that we reviewed that was published by Denisov and Beichman. Two of the questions that we began to ask were how does time become a factor in the stability of the solution and how much do we want to alter the boundaries of the patch. For the remainder of the week our team decided to alter the boundaries of the patch a few different ways. We started off with making the boundary of the solution really wiggle and made sure to have the left-most boundary -L+(1-Cos(ωy)) and the right-most boundary L+(1-Cos(ωy)). Omega is the angular frequency of the function and the very first alteration made sure to make ω=1. In this case, when analyzing the integral to attempt in finding parts of the velocity field u=(u1,u2), we were happy to come to the conclusion that u1 was 0. This great realization was great until we had realized that when looking at the integrand and plotting it, there was dependency on both x and y, since then it has been a challenge in finding patterns for which we can use to find u2.

After figuring that the previous attempt was a challenge we moved on to altering the boundaries of the path differently. We took a look at altering the boundary where instead of it oscillating too much, it would just be one bump. Therefore, we decided to make the boundaries -L+A(y^2+2πy) and L+A(y^2+2πy), where the amplitude, A, we looked at was ¼. Finally, we ran into challenges with this one as well and one next step we took was to look at a profile that is an even extension of a linear function of y. While we were juggling between profiles, Jenn let us know that we had to account for a variable very important in convolution. Therefore, next week we will take a look at the profiles to make sure all the variables needed are accounted for.

## Taking a Step Back

This week we worked on understanding in detail the behavior of one of the steady state solutions for Model 2s (the model where both producers and scroungers die linearly and the scroungers depend on the producers to eat):
P → ( r  ( ( k / β ) – δ_1) )  /  ( b k / β )        S → 0        F → δ_1 /  β
We found that:
1) There will never be a case in which the radical in the eigenvalue is a real number and the entire eigenvalue is negative.
2) For almost all cases the radical will be imaginary and therefore the entire eigenvalue will be negative
3) The only way that the radical would be real is if δ_1 is greater than (4 k^2 / β^2) / (r+4 k /β)
4) For the third eigenvalue to be negative, δ_1 >= k /β which is very unlikely to occur because β would have to be very small
5) It could be possible that the first two eigenvalues are positive if δ_1 >= k \ β
6) Our questions
– How are all three eigenvalues sometimes negative?

## Scroungers Report

This week consisted of making more models with the goal of finding one in which producers, scroungers and food coexist and are stable with the logic that:

1. Producers find food on their own
2. Scroungers eat the left overs of the producers or they run into food on their own (however, the latter is less likely)
3. Food has logistic growth and is eaten at a rate corresponding with the producers and scroungers

We’re moving towards simplifying the model. In doing this, we can take a step back and really try to understand the minute details of the models.

## Just like us, u_2 can do it!

This week we focused on deriving the velocity function u which we had realized was made of two components u=(u1,u2) . As we had learned before the first component u1=0 but we then figured out that the second component which began as a double integral was truly a single integral with an integrand with a natural log. This is what the second component of the velocity field looked like u2=Log{[Cosh(x-L)-Cos(y-t)]/[Cosh(x+L)-Cos(y-t)]}dt  and we attempted to solve the integral by using multiple methods of integration, and with no success, we decided to plot the integrand to gain an idea of the integral value. Fixing x and L and manipulating in y, we found that the Cos(y-t)term in the integrand did not have as large an effect as the Cosh(x-L)term. Simplifying the integrand even further with the expanded form of hyperbolic cosine, we were then able to analyze the behavior of the integrand for certain values of x within the bound [-L, L]. Using the range of these values we were able to create a piecewise function, u2(x)= {2L*2π, x <=-L}, {-2x*2π, -L< x < L}, {-2L*2π, x >=L}. Together, we found the vector plot of the velocity function which tells us that not only is our solution for the strip unique, it is also a steady state solution.

## Scroungers Progress Report #Scroungers

This week we worked on different ordinary differential equation models to help us understand the interactions between scroungers, producers and prey. In this process we learned how to interpret models by using Mathematica to solve for the steady state solutions of our models; we then use the Jacobian of the system of equations that make up each model to determine if the model is stable or unstable at the solution. We will continue to create more complex models to better understand the relationship between scroungers, producers and prey.

## Streaming through Week 2

Towards the beginning of the week, our team decided to take a closer look at some of the literature provided to us in order to understand some context to the problem we will be working on. We also looked at new functions and their hyperbolic behavior, Sinh[x] and Cosh[x]. These two functions were pivotal to our understanding of the Stream Function (Ψ(x,y) = Γ∗θ(x,y) =∫1/2log(cosh(x−ξ1)−cos(y−ξ2))θ(ξ1,ξ2)dξ1dξ2) and its relation to velocity and vorticity.

For the remainder of the week, our team took a further look at convolution and its relevance to the Stream Function. This process involved us taking several partial derivatives of the Stream Function with respect to different variables. Once we took a look at this, we decided to take the Laplacian of the Stream Function and it surprisingly yielded something that had some relation to the velocity u(x,y). Moreover, some of the literature that was provided demonstrated some u (velocity) that we are trying to prove is a steady state solution to the partial differential equation that we are working with. As of now we are continuing to prove that the function u(x,y) (velocity) is a steady state solution to the differential and we are doing so by using a Dirac function.

Part of the process this week was to gain a better understanding of what the three functions (Stream Function, velocity, and vorticity) mean in relation to each other and how to move forward in our project.

## Week 2: Starting Research Projects

Goals:

• Further practice with Mathematica
• Learn to use LaTeX for typing mathematics
• Start work on individual research problems

Our Research Teams:

• Pattern Formation in Animal Foraging (#scroungers) with Nessy Tania
• Caira Anderson
• Issa Susa
• Vortex Patch Dynamics (#vortexpatches) with Jen Beichman
• Victoria Camarena
• Sasha Shrouder

• Pattern Formation
• Vortex Patch Dynamics

## Our Ideas for Improving Smith College and Higher Education

Here is our ongoing list of ideas on how to improve the Sciences at Smith and in the world.

• Provide inter-student mentor options, or at least have lunch presentations/meetings with students where they address a lot of the insecurities people have about their skills and smarts and share information about how to navigate their new environments as college students, STEM students, and Smith students. Maybe along with presentation of major, a student panel in the beginning of the semester.

## Week 1: PDE and Mathematical Computing Minicourses

Goals:

• Learn basics of PDEs and computing with Mathematica
• Form research groups
• Write MathFest presentation abstracts

Mathematica Materials:

Mathematica Day 1: Mathematica Notebook

Mathematica Day 1: Worksheet

Mathematica Day 2: Solving ODEs and steady state solutions

Mathematica Day 3: Fourier Series Demo

Mathematica Day 3: Heat Separation