I am creating a nonlinear version of my 5 stage class model. I am also working on a toy model (1 dimensional) to test out different ways of representing density dependent death rates. I will be meeting with Heideman tomorrow so I will probably ask him more about parameters and make sure the changes I have made so far look alright. I will also be working on my abstract that is due Friday.

It has been awhile since I have posted and quite a few things have changed. After my presentation on Friday I got some good feedback on what to change/add. I am coding a new model where I still have three stage classes but now the Young Adults (the middle class) will no longer reproduce; they represent a dispersing group of mice. I will now be calling them the Weaned (W) stage class. So now I have Juvenile, Weaned, and Adult. I have also added two new stage classes that represent Adult females that are lactating or are pregnant (they cannot reproduce so they are separate from the Adult class)

I am also reading about logistic models and using them to incorporate a density dependent death rate.

Today was a little frustrating. I had thought I got my maturation rates down awhile ago, but when I was checking back on all my rates I discovered that the way I had calculated the maturation rates did not seem correct. I had manually calculated them (w help of matlab) using the info that mice sexually mature in about six weeks (42 days). Hopefully I can fix this soon, even though at the moment I am confused on how to go about doing so.

I also fixed my birth rates (for my linear model) and my seasonal issues after my talk with Heideman. I also keep getting confused on the way to find birth rates for my nonlinear model; my idea was to multiply a reproductive success rate with a meeting rate to give me a "birth rate" but I want to somehow make sure this corresponds with the birth rate info Heideman gave me (4 litters per year, each litter consists of about 4 mice).

I will probably take a break from parameters tomorrow and focus on getting a draft of my lab report done in tech.

I have been varying my maturation rate of the young adults to adults (I found data that suggested mice in warmer climates mature sexually faster) and I have found some interesting results when I differ the maturation rates between the P and NP(the NP population explodes, while the P seems not to). I also fixed a latex problem that I have been having for awhile.

Today I met with Professor Heideman and there are some important changes that I need to make to the model. He would like me to incorporate a kind of carrying capacity so that as the population increases there is a greater risk of dying (i.e less nutrients, fewer places to hide, etc.). He also told me that spring, summer, and fall should have similar parameters, while the winter's parameters differ (before I had assumed fall and winter were similar and summer and spring were similar). Next he would like me to add an intermediate group of mice, meaning that there would be a group of mice between P and NP, and that they have a probability of producing half the time in the winter. Adding this group will help with the genetic component of my model; when a NP and P mate there children would become intermediates (assuming heritability is 100%) .

Sorry I have not posted in awhile. I have put my "basic" nonlinear model into matlab and it is up and running. I have been trying to find birth rate terms in terms of copulation attempts per couple* pregnancy success rate. This birth rate needs to incorporate both males and females since my nonlinear model depends on both sexes (and it is the only way my units cancels out). I found a couple papers that may help with this kind of data. I also found some interesting information about maturation rates; mice in warmer climate mature faster than mice in colder climates, while the difference is only about a weeks time, it could impact my data, so I plan to play around with this in my matlab model. I am now going to work on developing a carrying capacity (I need to decide what I want it to depend upon i.e nutrients, rainfall, etc.) A critical depending factor for this would include the diet of white footed mice; I found that the main sources of food throughout the year are insects, but additionally in the spring and summer they eat fruit and in the winter and fall they eat seeds/green vegetation.

Now that I have got my linear model down, I have coupled my two populations (photo and non-photosensitive) together and created a basic nonlinear model. In this model all age classes are split into NP and P, and the Adult class is also split into male and female. A lot of new parameters arise from this, however some of these may collapse after I talk to Prof Heideman. Right now I am working on creating all my equations and making a parameter/variable table in Tech. Once I have this done (it may take awhile), I will start putting this model into matlab to analyze it. Later on I will add nonlinear functions to make additional models, for example my birth rate may become a function of both the number of NP and P and nutrients.

I realized yesterday that my matlab code for my linear model was not accurately representing my model; when I made all the birth rates zero the Juvenile population never died off, which is what should have occured. The problem has now been corrected (my itterations were not properly working) and my graphs look quite different from the ones I had showed Meagan and Prof Day at our meeting on Wed. (Prof. Day I can email you my new pictures!). I looked at the stable age distribution and the dominant eigenvalue and found that the population will reach an unstable steady state and explode (this was conducted by using my parameters I had found in papers).

Today I found a new paper with a lot of the parameters explicitly written in it; this paper also includes various parameters according to population density, so this may help when I begin to add nonlinear terms.

I also looked at the distriubution of seasons according to the Coordinated Universal Time (UTC), and I found that each seasons begins or ends with a solstice or equinox and lasts for approximately 13 weeks. I think that the seasonal distributions I used in my model (each 13 weeks) is a good estimate for the time being.

I plan to play around with my linear model a little more and then I will add a nonlinear term and try to code that next.

I am currently putting my linear model into matlab; I have discritized my model and have decided to use a time scale of weeks (this may change to days if it looks too choppy when I graph my model). The linear model will be divided into a non-photosensitive and a photosensitive group, and these groups will not interact for the moment; I am doing this so that when I run simulations on each individual population I will be able to check that my code is working properly (e.g for the photosensitive group the Juvenile population will fluctuate in a sinusoidal fashion) I have also created a loop so I can vary the birth rates seasonally.

Today I also spent some time researching parameters. Finding birth rates for separate seasons is difficult, but I am able to find average birth rates, so I may have to make some estimates/talk to Prof Heideman.

I went to the research ethics lecture today led by Prof Heideman. While some of the scenarios in the talk would not apply to math research students, the issues concerning citation and credit were helpful. It was also interesting to see some of the barriers (grants, patents, etc.) that come up during scientific/laboratory research .

I am now trying to take my linear model and make it nonlinear by determining which of my paramters should depend on the environment, population size, etc. I am creating a couple different models with different assumptions. My first model at the moment only has birth rate dependent on the environment, and my second model has both birth and death rate depending on the environment. I am at the moment wondering weather I should some how add a carrying capacity that would decrease birth rate and increase death rate when the population becomes too large. An example of this is represented in the equation below, where the change in population size is dependent on the population size N, the growth rate r, and the carrying capacity K (if N=K, then the population size will remain unchanged).

I also read about the Ricker function today (another function that could help incorporate density dependence). I am trying to decide what would be a better fit for the model.

dN/dt = B(N)N − dN where B(N) is a birth rate function and d is the death rate (not dependent on anything). B(N) can be written as a Ricker function, where B(N) = b*e^(−N), with b > d, where b is the birth rate. Here also when N, the population size becomes very large the birth function gets close to zero.

New Linear Model

b = birth rate from Adults

s = birth rate from Juveniles

l = birth rate from Adults and Juveniles

m = maturation rate of Young

k = maturation rate of Juvenile

d = mortality rate of Young

q = mortality rate of Juvenile

r = mortality rate of Adult

After meeting with Prof Heideman I have come up with this linear model. Now Juveniles are all mice babies from birth till weening (approx 3 weeks), Young Adult are mice that are potentially reproductive, and Adults are fully reproductive. This model does not represent NP vs. P or male vs. female.

I am going to modify my equations to show which parameters will possibly become functions dependent on other factors. I am also now most likely going to discretize my model (perhaps examining only two seasons: winter and summer and working up to four seasons).

I completed the Routh Hurwitz criteria for my 3x3 matrix a couple of days ago. It was difficult to extrapolate any relevant information from this; this is probably due to the fact that I have quite a number of parameters/the model used was linear/I looked at the stability of the trivial steady state.

I had a meeting with Prof Heideman today which was extremely helpful for the development of my model. Some previous notions have changed, and my model will look different (I will post a new diagram soon). I got a better handle on the meaning and approximate values for many of my parameters; I will also look at some journal articles on Peromyscus to get some more information for my parameters. I plan to continue to meet with Prof Heideman during this summer so that I can make the most representable model.

My plan is to redo my linear model to meet Prof Heideman's expectations and then identify the parameters that need to be changed from a constant to a function dependent on variables such as environment, etc. I plan to start experimenting with matlab shortly.

Lastly, today was our last GIS class; hopefully in the future I will use some of what I have learned in the spatial aspect of my model.

Our second GIS meeting was today, and I learned how to work with vector data (which consists of points, lines, and polygons) and how to we can modify such data to fit our needs. We can put this kind of data on top of maps/images to represent whatever we want. Pretty cool stuff.

I have downloaded Latex and will continue to play with it so I get more comfortable with the language. I have unsuccessfully downloaded matlab, and IT is also currently stooped!

While I did not get to do much modeling today, I got some good ideas on where to take my model. I plan on making my linear model nonlinear and writing out the explicit quadratic terms. I also plan on programming my linear model and just doing a couple of simulations (I will probably change my model from continuous to discrete), and hopefully I will try and do a bifurcation diagram for this model, (although this might not be interesting since I will be doing it for a linear model). My goal is that once I talk to Heideman and understand my parameters more clearly that I can build and a nonlinear model and program that.

Basic Model:

Parameters:

Sorry for starting this blog so late! This first week has been pretty productive. I took the model that I had been using and temporarily changed it (I assumed there are only two photosensitive groups instead of a continuum) and diagrammed it to look at the parameters that could possibly come up.

By doing this I realized (with Meagan's help) that there are some more questions I should ask Prof Heideman, i.e. do maturation rates for young->juvenile differ for males vs. females or for photosensitive vs. nonphotosensitive. So I am going to meet with Prof Heideman on Tuesday morning, I may even ask him if I can see the mice!

I also read about a chapter and a half of Dynamic Models in Biology; its been pretty helpful in reviewing some key concepts in modeling. I checked out a couple of other books from the library and read a little bit of the Chaos book.

Meagan taught me how to use the Routh Hurwitz criteria , a way to check for stability for steady states. So after looking at an example, I tried it on my own model (my initial 3 dimension linear ODE); I still have to multiply a large thing out, but it's almost there!

Lastly, today we had our GIS class which was very cool and different, I think this class could eventually help when I need to add my spatial component to my model.