# Liora's Project
## Notes from meeting (8/21)
* we changed the arrow on the interactions diagram (now it goes from medusa to small fish --> this affects the derivative function of the small fish)
* we finished the interactions diagram
* we began making tables on here
## Next Meeting Agenda
* final checkover of functions
* finalize title
* go over subheadings for intro
* begin coding
* ****
* ~~**Define new schedule**~~
## Voluntary tasks for next meeting
* Send functions (Liora).
* Update diagram (Liora).
* Review and comment functions (Carlos).
* **Regeneron**
* Write Regeneron recommendation (Carlos). **(POSTPONED)**
## State variables table
| Variable | Symbol | Description |
| -------- | -------- | -------- |
| Polyp | $$P$$ |Polyp Population |
|Ephyra| $$E$$| Ephyra Population
|Medusa|$$M$$|Medua Population
|Plankton|$$PL$$|Plankton Population
|Small Fish|$$F_S$$|Small Fish Population
|Larvae|$$F_L$$|Larvae Population
|Egg|$$F_E$$|Fish Egg Population
|Adult Fish|$$F_A$$|Adult Fish Population
## Parameter table
| Parameter | Symbol | Description |
| -------- | -------- | -------- |
| Predation rate | $$\alpha_{b,a}$$ | Predation rate of species a on species b |
| Transition Rate | $$\eta_{j,i}$$ | Rate at which i becomes j |
| Death Rate | $$\delta$$ | $$F_a (\delta_{F_a}+\lambda)$$ with $\lambda=$ Fishing rate |
| Efficiency Rate| $$\epsilon_i$$ | Effect of species j consuming species i. Which means, how many of species j is produced per unit of consumed species i |
### Regeneron important dates (and docs)
* Project is due November 9th.
* Our own date is October 16th (Everything but conclusions must be ready).
* Methods (Carlos and Liora) **Sept 4th**:
* Diagram (Liora) **Ago 28th**.
* Functions () (solved to equilibrium?) **Sept 4th** (Carlos and Liora).
* Code (Liora and Carlos) **Sept 4th**.
* Debugging (Liora and Carlos) **Sept 11th**
* Introduction (Liora and Carlos) **SET DATE (Liora)**.
* Results (Liora) **Sept 25th [tops]**.
* Discussion (Liora and Carlos) **TBD**.
* Conclusions (Liora and Carlos) **TBD**.
* Ms. York (will supervise the process).
$$P$$
$$M_i \in \{p, e, m\} => M_p, M_e, M_m$$
$$F_j \in \{h, l, s, a\} => F_h, F_l, F_s, F_a$$
$$ \alpha_{M_p F_s} $$
# Chart
## Key for Chart
| Number | Symbol | Description |
| -------- | -------- | -------- |
| 1 + 5 + 6 + 17 + 18 + 19 | i = small fish population, j = adult fish population, l = medusa population, $$g \in \{ polyp, ephyra, medusa ,plankton\}$$| $$\frac{dF_j}{dt}=\eta_{ji} F_i (1-\frac{F_j+\alpha_{F_jJ_l}J_l}{K_j})-\delta_{F_j}F_j-F_j\sum_{g}\alpha_{jg}J_g$$ |
| 1 + 5 + 6 + 17 + 18 + 19 | i = small fish population, j = adult fish population, l = medusa population, $$g \in \{ polyp, ephyra, medusa ,plankton\}$$| $$\frac{dJ_j}{dt}=\eta_{ji} J_i (1-\frac{J_j+\alpha_{J_jF_l}F_l}{K_j})-\delta_{J_j}J_j-J_j\sum_{g}\alpha_{jg}F_g$$ |
| | $$g \in \{ polyp, ephyra, medusa ,plankton \}$$ c = saturation constant $$\eta_{ji_{max}}$$ = the max rate at which stage i transitions to stage j |$$\eta_{ji}= \eta_{ji_{max}} e^{ \left( \frac{(\epsilon_iF_i\sum_{g} \alpha_{gi} J_g) -c}{\epsilon_iF_i\sum_{g} \alpha_{gi} J_g +c } +1\right)}$$ |
| 7 + 8 + 17 | i = ephyra population, j = medusa population, l = adult fish population|$$\frac{dF_j}{dt}=\eta_{ji} F_i (1-\frac{F_j+\alpha_{F_jJ_l}J_l}{K_j})-\delta_{F_j}F_j$$ |
| 1 + 2 + 12 + 13 | i = larvae population, j = small fish population, l = adult fish population, $$ g \in\{m,n \}$$ |$$\frac{dF_j}{dt}=\eta_{ji} F_i(1-\frac{F_j}{Kj})-\eta_{lj}F_j-F_j\sum_{g}\alpha_{jg}J_g$$ |
|3 + 4 + 20 |i = adult fish population, j = egg population, h = larvae population, b = medusa population | $$\frac{dF_j}{dt}=\eta_{ji} F_i(1-\frac{F_j}{Kj})-\eta_{hj}F_j-\alpha_{jb}F_jJ_b$$ |
|2 + 3 + 21 |i = egg population, j = larvae population, s = small fish population, b = medusa population|$$\frac{dF_j}{dt}=\eta_{ji} F_i(1-\frac{F_j}{Kj})-\eta_{sj}F_j-\alpha_{jb}F_jJ_b$$ |
|9 + 10 + 11 + 12 + 14 + 19 |i = medusa population, j = polyp population, e = ephyra population population $$ g \in\{m,p \}$$|$$\frac{dF_j}{dt}=\sum_{g}\eta_{ji}F_i(1-\frac{F_j}{Kj})-\eta_{ej}F_j$$|
|8 + 9 + 13 + 15 + 18 |i = polyp population, j = ephyra population| $$\frac{dF_j}{dt}=\sum_{g}\eta_{ji}F_i(1-\frac{F_j}{Kj})-\eta_{bj}F_j$$ |
## General Equations:
- $$\frac{dF_j}{dt}=(\sum_{i}\eta_{ji}F_i)(1-\frac{F_j +\sum_l \beta_{jl} J_l}{Kj})-\eta_{bj}F_j-F_j\sum_{g}\sigma_{jg}J_g$$
- $$\frac{dJ_j}{dt}=(\sum_{i}\gamma_{ji}J_i)(1-\frac{J_j+\sum_l \beta_{jl} F_l}{Bj})-\gamma_{bj}J_j-J_j\sum_{g}\alpha_{jg}F_g$$
# Template model
```r
# logistic ####
# predict jellyfish population
library(deSolve)
Parms=list(a=15,e=1.5,K=30000,r2=1.5)
State=c(N=10000, R=30000)
Times=seq(from=0,to=5,by=0.1)
logistic=function(Time,State,Parms){# a,e,K,r2,N,R
# resource ####
# small fish
Parms_t=unlist(Parms)
a=Parms_t[1]
e=Parms_t[2]
K=Parms_t[3]
r2=Parms_t[4]
R=State[2]
N=State[1]
R=r2*R*(1-R/K)-a*N*R # r2 is growth of the resource, R is the resource population
# K is carrying capacity, a is the consumption rate of the consumer, N is consumer population
# consumer ####
# jellyfish
N=a*e*R*N*(1-N/R) # a is the consumption rate of the consumer, e is the efficiency rate
# N is the concumer population, R is the resource population
return(list(c(N=N,R=R)))
}
sim=ode(y=State,times=Times,func=logistic,parms=Parms)
plot(x=sim[,1],y=sim[,3],col="blue",type="l")
plot(x=sim[,1],y=sim[,2],col="red",type="l")
```
# New code version
```
# logistic ####
# predict jellyfish population
library(deSolve)
State=c(initialpopsvalues)
Times=seq(from=0,to=10,by=0.1)
parameters=list(
k = ,# Carrying capacity -> vector
eta , # rate at which fish stage i becomes j -> Matrix
gamma_ = ,# rate at which jellyfish stage i becomes j -> matrix
alpha = , # Predation fish over jellyfish -> matrix
sigma = , # Predation jellyfish over fish -> matrix
constan = , # Saturation constant -> scalar
beta_ = , # Competition coefficient between fish and jellyfish -> matrix
efficiency = # efficiency (rate of biomass conversion) -> scalar
)
logistic=function(Time,State,Parms){# a,e,K,r2,N,R
# resource ####
# small fish
Parms_t=unlist(Parms)
a=Parms_t[1] # call each variable a word
e=Parms_t[2]
K=Parms_t[3]
r2=Parms_t[4]
R=State[2]
N=State[1]
dFdt= (eta%*%F_i) * (1 - (F_i+ beta_%*%J_i)/(K_j)) - # new individuals per stage in Fish population
eta%*%F_i - # Individuals leaving stage i
F_i*(sigma_%*%J_i) # Number of fish that are consumed by any jelly fish stage
# dJdt =
outcome = c(dFdt, dJdt)
names(dFdt)<-c('names for each stage')
return(list(c(N=N,R=R)))
}
sim=ode(y=State,times=Times,func=logistic,parms=parameters)
```
**we are assuming that efficiency is the same
```
library(deSolve)
library(deldir)
library(tidyverse)
library(tensor)
library(deldir)
library(MASS)
library(abind)
library(future)
library(purrr)
library(furrr)
library(parallel)
library(tictoc)
library(future.apply)
trabajadores=detectCores()-1
source("Liora_Code2.R")
source("parallelparameters_Liora.R")
###############Parameter tibble #############
# parameter_tibble = # This line inputs the parameters for estimating other parameters for Liora model
# expand_grid(
# efficiencyf=0.9*c(0.1, 5.5, 10),
# efficiencyj=0.05*c(0.1, 5.5, 10),
# competitionfactorf=c(0.1, 5.5, 10),
# competitionfactorj=c(0.1, 5.5, 10),
# constant=1*c(0.1, 5.5, 10),
# k=1000*c(0.1, 5.5, 10),
# b=1000*c(0.1, 5.5, 10),
# predation_rate_factorf=c(0.1, 5.5, 10),
# predation_rate_factorj=c(0.1, 5.5, 10)
# )
parameter_tibble = # This line inputs the parameters for estimating other parameters for Liora model
expand_grid(
efficiencyf=0.9,
efficiencyj=0.005,
competitionfactorf=seq(0, 1, 0.25),
competitionfactorj=seq(0.001, 0.08, 0.02),
constant=0.1,
k=1000,
b=1000,
predation_rate_factorf=seq(from=0.4, to=0.9, by=0.1),
predation_rate_factorj=seq(from=0.05, to=0.1, by=0.02)
)
#############################################
#######################################################################################################
# efficiencyf_=efficiencyf,
# efficiencyj_=efficiencyj_,
# competitionfactorf_=competitionfactorf_,
# competitionfactorj_=competitionfactorj,
# constant_=constant,
# k_=k,
# b_=b,
# predation_rate_factorf_=predation_rate_factorf,
# predation_rate_factorj_=predation_rate_factorj,
# transitionf=Fish, # matrix
# transitionj=Jellyfish,
# predation_fish_=predation_fish,
# predation_jellyfish_=predation_jellyfish,
# competition_fish_to_medusa_=competition_fish_to_medusa,
# competition_medusa_to_fish_=competition_medusa_to_fish
#######################################################################################################
sim =
function(efficiencyf_=efficiencyf,
efficiencyj_=efficiencyj,
competitionfactorf_=competitionfactorf,
competitionfactorj_=competitionfactorj,
constant_=constant,
k_=k,
b_=b,
predation_rate_factorf_=predation_rate_factorf,
predation_rate_factorj_=predation_rate_factorj,
state, t){
settings=parallel_parameters(efficiencyf_=efficiencyf_,
efficiencyj_=efficiencyj_,
competitionfactorf_=competitionfactorf_,
competitionfactorj_=competitionfactorj_,
constant_=constant_,
k_=k_,
b_=b_,
predation_rate_factorf_=predation_rate_factorf_,
predation_rate_factorj_=predation_rate_factorj_
)
Params = settings$parameters
State=settings$State
out = ode(y = State, times = times, func = logistic, parms=Params)
res_df =
as.data.frame(out) |>
as_tibble()
#names(res_df) = names_df$name
return(list(outcome = res_df, parameters = Params))# as_tibble(
}
times <- seq(0, 0.5, by =0.1)
results =
pmap(
.l = parameter_tibble,
.f = sim,
t = timeseq
)
source("sortingresults.R")
df=as_tibble(bind_rows(lapply(results, results_sorting)))
dtf=df |>
mutate(totalfish=rowSums(dplyr::select(df, contains('fish'))))|>
mutate(totalmedusa=rowSums(dplyr::select(df, c("polyp", "ephyra", "medusa"))))
plot_total_fish =
dtf |>
#filter(totalfish>10)|>
mutate(competitionfactor_ratio=as.factor(competitionfactorf/competitionfactorj))|>
ggplot(aes(predation_rate_factorj,totalfish, group = predation_rate_factorf, color = as.factor(predation_rate_factorf))) +
geom_line() +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1)) +
facet_grid(competitionfactorf ~ competitionfactorj, labeller = label_both)
plot_total_fish |>
show()
```
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