### Example Matrix Population Model ### ## Parameters ## T = s.1 s.2 b = 20 #Model will run for 20 time steps plus time zero = 0.5 #Survival from larva to pupa = 0.4 #Survival from pupa to adult 5 #Birth rate ## Define vectors and initial conditions ## n = matrix(0,3,T+1) n[1,1] = 5 #Larva vector starts at a pop of five n[2,1] = 5 #Pupa vector starts at a pop of five n[3,1] = 5 #Adult vector starts at a pop of five ## Begin for loop ## for (t in 1:T) n[1,t+1] n[2,t+1] n[3,t+1] } { = b*n[3,t] = s.1*n[1,t] = s.2*n[2,t] ## Plot the model ## plot(seq(0,T,by=1),n[1,],type='b',xlab='Time (days)',ylab='Population',col='red',ylim=c(0,30)) points(seq(0,T,by=1),n[2,],type='b',col='blue') points(seq(0,T,by=1),n[3,],type='b',col='green') legend('topright',c('Larvae','Pupae','Adults'),lty=c(1,1), lwd=c(2.5,2.5), col=c('red','blue','green')) Math in the City Intro to Matrix Population Models Homework 2 Due 9/6 1 Introduction Model and plot the following in R. The yellow fever mosquito, Aedes aegypti, is well known for its role as a vector for tropical fevers including dengue and Zika, and, as such, is often the target for control efforts. Before modeling the effectiveness of control efforts, we must first model population growth. As with many insects, matrix population models are a useful way of modeling life stages. Mosquitos are holometabolous, meaning they have four life stages: eggs, larvae, pupae and adults. In addition, adults females go through a pre-blood and gonadotropic stages where they feed on blood before laying eggs (on average four gonadotropic cycles). Larvae and pupae can, for our purposes, be grouped into a single aquatic category. Experimental results yield the following approximates. Value Time as Egg Time as Aquatic Time as Pre-Blood Time for each Gon. Stage Birth Rate Aquatic Survival Rate Adult Stages Survival Rate Measurement 5 days 10 days 1.4 days 2.5 days 42 eggs per female per cycle 60% survive the aquatic stage 90% survive a single adult stage Let the time scale for t be in days. Let the state or population vector represent the following: m1 eggs m2 aquatics m3 pre-bloods m4 gon. stage 1 m = := m gon. stage 2 5 m6 gon. stage 3 m7 gon. stage 4 m8 final eggs and death To determine the probabilistic parameter ri := likelihood to transition from stage i to i + 1, 1 . To calculate the birth rate, b for the model, we we should simply let ri = Time in stage i must divide the birth rate by 2 since only females lay eggs. Perhaps we should also divide by time spent in each gonadotropic cycle, since females only lay once per cycle too, huh? Survival probabilities (s` for larval survival and sa for adult survival) just fall right out, but you should think about why that translates to sri i in the model. 1 2 Model c r1 0 0 0 b b b r1 r2c sr` 2 0 0 0 0 0 0 r2 sr2 r3c sra3 0 0 0 0 ` r3 c r4 0 0 r3 sa r4 sa 0 0 0 mt = r4 c r5 0 0 0 r s r s 0 0 4 a 5 a r5 c r6 0 0 0 0 r5 sa r6 sa 0 c r7 r6 0 0 0 0 0 r6 sa r7 sa 0 0 0 0 0 0 r7 sra7 3 b 0 0 0 mt1 0 0 0 0 Assignment Create a vector for each population stage that is the length of Max-Time + 1. You may choose how many days of population growth you would like to simulate in your model (it would be best to let MaxTime be a parameter at the start of your model so that you can start it at 1 or 2 and make it bigger once your model is fully programed and debugged). You may also choose the initial conditions of your state/population vector (just not all 0, because that would be boring). Model the state equation over time using a for loop. Plot the results for every stage in one large image using either the \"par\" or \"points\" commands in R (one allows your to join multiple plots, while the other allows you to plot multiple vectors on top of the same image even including a legend). Be sure to comment using \"#\" throughout your work, so that it is possible for others to follow. Note that in the above model, parameters with the superscript c are the probabilistic complement. [1] Dawson et al.. Population effects of malathion to Aedes aegypti vary with temperature and age of exposure. Unpublished, 2017. 2