Evolution Models

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knitr::opts_chunk$set(echo = TRUE)

library(ecoevoapps)
library(kableExtra)
library(tidyr)
library(ggplot2)
ggplot2::theme_set(ecoevoapps::theme_apps())

Bu aplikasyon avcı ve av etkilşimlerini incelemektedir. Avcı ve av modellerinin farklı versiyonları olsa da (aşağıdaki sayfalarda bu versiyonlara ulaşabilirsiniz), hepsi genel bir iskelet paylaşır. Av türü avcının yokluğunda üstel ya da lojistik olarak büyüyen bir popülasyondur. Avcı tüm enerjisi icin av türüne bağımlıdır ve eğer sistemde hiç av yok ise avcı popülasyonu tükenecektir. Av popülasyonunun nüfusu büyüdükçe, avcı popülasyonu da büyümektedir. Fakat, avcı popülasyonu büyüdükçe, av popülasyonu avlanmayla beraber küçülür. Bu etkileşim türü çoğu avcı-av modelinde gözlemlediğimiz periyodik döngüye yol açar.

Tarih boyunca, avcı-av modellerinin versiyonları parazitler ve konakları arasındaki dinamikleri çalışmak icin de kullanılmıştır. Dolayısıyla, burada “av” türünü $H$ degişkeni ve “avcı” türünü $P$ değişkeni ile gösteriyoruz.

{.tabset}

Lotka-Volterra Avcı-Av modeli

Klasik Lotka-Volterra avcı-av modeli üstel büyüyen bir avı ve tükenme noktasına erişmeden tüketen bir avcıyı modeller (“Tip 1 fonksiyonel yanıt”, bakınız):

[ \begin{align} \frac{dH}{dt} &= rH - aHP\ \ \frac{dP}{dt} &= eaHP - dP \end{align} ]

pars_vars <- c("$H$", 
               "$P$", 
               "$r$", 
               "$a$", 
               "$e$", 
               "$d$")
descriptions <- c("Population size of the prey",
                 "Population size of the predator",
                 "Per capita growth rate of the prey",
                 "Attack rate of the predator",
                 "Conversion efficiency of the predator",
                 "Death rate of the predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

Sabit durum eşegim çizgileri için denklemler: [ \begin{align} P^ &= \frac{r}{a}\ \ H^ &= \frac{d}{ea} \end{align} ]

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    ## r, a, T_h, e, d

    sliderInput("r_lv_pred1", label = "Per capita growth rate of Prey", 
                min = .0001, max = 1.0, value = .5),
    sliderInput("a_lv_pred1", label = "Predator attack rate", min = .001, 
                max = 1.0, value = .1),
    sliderInput("e_lv_pred1", label = "Predator conversion efficiency",
                min = .001, max = 1.0, value = 0.2),
    sliderInput("d_lv_pred1", label = "Per capita death rate of Predator",
                min = .0001, max = 1.0, value = .3),

    ### Ask users for initial conditions -----
    #N1, N2

    numericInput("H_lv_pred1", label = "Initial population size of Prey", 
                 min = 1, value = 10),
    numericInput("P_lv_pred1", label = "Initial population size of Predator", 
                 min = 1, value = 10),

    ### Ask users for time to simulate ----
    numericInput("t_lv_pred1", label = "Timesteps", min = 10, value = 100),

    checkboxGroupInput("vectors_lv_pred1", label = "Display vector field?", 
                       choices = c("Yes" = "Yes"), selected = "Yes")

  ),  


  mainPanel(renderPlot(plot_lvpred1()),
            renderPlot(np_lvpred1())
            )

)

# Set the initial population sizes
init_lv_pred1 <- reactive({c(H = input$H_lv_pred1, P = input$P_lv_pred1)})

# Set the parameter values
pars_lv_pred1 <- reactive({
  c(r = input$r_lv_pred1, a = input$a_lv_pred1, 
    e = input$e_lv_pred1, d = input$d_lv_pred1)
})

# Time over which to simulate model dynamics
time_lv_pred1 <- reactive({seq(0, input$t_lv_pred1, by = .1)})

# simulate model dynamics
out_lv_pred1 <- reactive({
  data.frame(run_predprey_model(
    time = time_lv_pred1(),
    init = init_lv_pred1(),
    params = pars_lv_pred1()
  ))
})

# Plots ------
## make abundance thru time plot
plot_lvpred1 <- reactive({
  plot_predprey_time(out_lv_pred1())
})

# plot trajectory of population
np_lvpred1 <- reactive({
  if("Yes" %in% input$vectors_lv_pred1) {
    plot_predprey_portrait(out_lv_pred1(), pars_lv_pred1(), 
                                         vectors_field = TRUE)
  } else {
    plot_predprey_portrait(out_lv_pred1(), pars_lv_pred1(), 
                             vectors_field = F)
  }

})

Lojistik Av

Bu model klasik Lotka-Volterra modelinin biyolojik gerçekliğini arttıracak varsayımlar ekler. Özellikle, klasik modelde, av popülasyonunun üstel büyümesi varsayımı rahatlatılır ve av taşıma kapasitesi $K$ ile lojistik büyür.

[ \begin{align} \frac{dH}{dt} &= rH \biggl(1-\frac{H}{K}\biggr) - aHP \ \ \frac{dP}{dt} &= eaHP - dP \end{align} ]

pars_vars <- c("$H$", 
               "$P$", 
               "$r$", 
               "$K$", 
               "$a$", 
               "$e$", 
               "$d$")
descriptions <- c("Population size of the prey",
                 "Population size of predator",
                 "Per capita growth rate of the prey",
                 "Carrying capacity of the prey",
                 "Attack rate of predator",
                 "Conversion efficiency of predator",
                 "Death rate of predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

Sabit durum eşegim çizgileri için denklemler:

[ \begin{align} P^ &= \frac{r}{a}\bigg(1 - \frac{H}{K}\bigg)\ \ H^ &= \frac{d}{ea} \ \end{align} ]

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    ## r, a, e, d, K

    sliderInput("r_logprey", label = "Per capita growth rate of Prey",
                min = .0001, max = 1.0, value = .5),
    numericInput("K_logprey", label = "Carrying capacity of Prey", 
                 min = 1, value = 100),
    sliderInput("a_logprey", label = "Predator attack rate",
                min = .001, max = 1.0, value = .1),
    sliderInput("e_logprey", label = "Predator conversion efficiency", 
                min = .001, max = 1.0, value = 0.5),
    sliderInput("d_logprey", label = "Per capita death rate of Predator",
                min = .0001, max = 1.0, value = .3),



    ### Ask users for initial conditions -----
    numericInput("H_logprey", label = "Initial population size of Prey",
                 min = 1, value = 10),
    numericInput("P_logprey", label = "Initial population size of Predator", 
                 min = 1, value = 10),

    ### Ask users for time to simulate ----
    numericInput("t_logprey", label = "Timesteps", min = 1, value = 100),

    checkboxGroupInput("vectors_logprey", label = "Display vector field?", 
                       choices = c("Yes" = "Yes"), selected = "Yes")

  ),  


  mainPanel(renderPlot(plot_logprey()), 
            renderPlot(np_logprey())
            )
)

# Set the initial population sizes
init_logprey <- reactive({
  c(H = input$H_logprey, P = input$P_logprey)
})

# Set the parameter values
pars_logprey <- reactive({
  c(r = input$r_logprey, K = input$K_logprey, 
    a = input$a_logprey, e = input$e_logprey, d = input$d_logprey)
  })

# Time over which to simulate model dynamics
time_logprey <- reactive({seq(0, input$t_logprey, by = .1)})

# Simulate the dynamics
out_logprey <- reactive({
    data.frame(run_predprey_model(
    time = time_logprey(),
    init = init_logprey(),
    params = pars_logprey()
  ))
})

# Plots ------

## make abundance thru time plot
plot_logprey <- reactive({
  plot_predprey_time(out_logprey())
})


np_logprey <- reactive({
  if("Yes" %in% input$vectors_logprey) {
    plot_predprey_portrait(out_logprey(), pars_logprey(), 
                                         vectors_field = TRUE)
  } else {
    plot_predprey_portrait(out_logprey(), pars_logprey(), 
                             vectors_field = FALSE)
  }
})

Tip 2 Fonksiyonel Yanit

Bu model avın üstel büyümesine geri döner, fakat av türünün varsayımları acısından biyolojik gerçekliği arttırmaya yönelir. Klasik L-V modelinde avcı-avı yakalama/tüketme sürecinde zaman harcamaz ve hiç bir zaman doyma noktasına erişmez. Bu modelde avcı türü tip 2 fonksiyonel yanıt gösterir. Bu avcının avı tüketme sürecindeki yeterliliğinin zamanla kısıtlıdır.

[ \begin{align} \frac{dH}{dt} &= rH - \frac{aH}{1+aT_hH} P\ \ \frac{dP}{dt} &= eP \frac{aH}{1+aT_hH} - dP \end{align} ]

pars_vars <- c("$H$", 
               "$P$", 
               "$r$", 
               "$a$", 
               "$T_{h}$", 
               "$e$", 
               "$d$")
descriptions <- c("Population size of the prey",
                 "Population size of the predator",
                 "Per capita growth rate of the prey",
                 "Attack rate of the predator",
                 "Handling time of the predator",
                 "Conversion efficiency of the predator",
                 "Death rate of the predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

Sabit durum eşegim çizgileri için denklemler:

[ \begin{align} P^ &= \frac{r}{a} +rT_hH\ \ H^ &= \frac{d}{ea - adT_h} \ \end{align} ]

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    ## r, a, T_h, e, d
    sliderInput("r_lv_pred2", label = "Per capita growth rate of Prey",
                min = .0001, max = 1.0, value = .5),
    sliderInput("a_lv_pred2", label = "Predator attack rate",
                min = .001, max = 1.0, value = .1),
    sliderInput("T_h_lv_pred2", label = "Predator handling time",
                min = 0.001, max = 1.0, value = 0.2),
    sliderInput("e_lv_pred2", label = "Predator conversion efficiency", 
                min = .001, max = 1.0, value = 0.7),
    sliderInput("d_lv_pred2", label = "Per capita death rate of Predator",
                min = .0001, max = 1.0, value = .3),

    ### Ask users for initial conditions -----
    numericInput("H_lv_pred2", label = "Initial population size of Prey",
                 min = 1, value = 10),
    numericInput("P_lv_pred2", label = "Initial population size of Predator",
                 min = 1, value = 10),

    ### Ask users for time to simulate ----    
    numericInput("t_lv_pred2", label = "Timesteps", min = 1, value = 100),

    checkboxGroupInput("vectors_lv_pred2", label = "Display vector field?", 
                       choices = c("Yes" = "Yes"), selected = "Yes")

  ),  


  mainPanel(renderPlot(plot_lvpred2()),
            renderPlot(np_lvpred2())
  )
)

# Set the initial population sizes
init_lv_pred2 <- reactive({
  c(H = input$H_lv_pred2 , P = input$P_lv_pred2)
})

# Set the parameter values
pars_lv_pred2 <- reactive({
  c(r = input$r_lv_pred2, a = input$a_lv_pred2, 
    T_h = input$T_h_lv_pred2 , e = input$e_lv_pred2, 
    d = input$d_lv_pred2 )})

# Time over which to simulate model dynamics
time_lv_pred2 <- reactive({seq(0,input$t_lv_pred2,by = .1)})

# Simulate the dynamics
out_lv_pred2 <- reactive({
  data.frame(run_predprey_model(
    time = time_lv_pred2(),
    init = init_lv_pred2(),
    params = pars_lv_pred2()
  ))
})


# Plots ------

## make abundance thru time plot
plot_lvpred2 <- reactive({
  plot_predprey_time(out_lv_pred2())
})

np_lvpred2 <- reactive({
  if("Yes" %in% input$vectors_lv_pred2) {
    plot_predprey_portrait(out_lv_pred2(), pars_lv_pred2(), 
                                         vectors_field = TRUE)
  } else {
    plot_predprey_portrait(out_lv_pred2(), pars_lv_pred2(), 
                             vectors_field = FALSE)
  }
})

Rosenzweig-MacArthur Modeli

Son olarak, Rosenzweig-MacArthur modeli lojistik büyüyen av ve tip 2 fonksiyonel yanıt gösteren avcıyı bir araya getirir.

[ \begin{align} \frac{dH}{dt} &= rH \biggl(1-\frac{H}{K}\biggr) - \frac{aHP}{1+aT_hH} \ \ \frac{dP}{dt} &= e \frac{aHP}{1+aT_hH} - dP \end{align} ]

pars_vars <- c("$H$", 
               "$P$", 
               "$r$", 
               "$K$", 
               "$a$", 
               "$T_{h}$", 
               "$e$", 
               "$d$")
descriptions <- c("Population size of the prey",
                 "Population size of the predator",
                 "Per capita growth rate of the prey",
                 "Carrying capacity of the prey",
                 "Attack rate of the predator",
                 "Handling time of the predator",
                 "Conversion efficiency of the predator",
                 "Death rate of the predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

Sabit durum eşegim çizgileri için denklemler:

[ \begin{align} P^ &= \frac{r}{a}\bigg(1 - \frac{H}{K}\bigg)(1 + aT_hH)\ \ H^ &= \frac{d}{ea - adT_h} \ \end{align} ]

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    ## r, a, T_h, e, d

    sliderInput("r_rm", label = "Per capita growth rate of Prey", 
                min = .0001, max = 1.0, value = .2),
    numericInput("K_rm", label = "Prey carrying capacity",
                 min = 1, value = 150),
    sliderInput("a_rm", label = "Predator attack rate",
                min = .001, max = 1.0, value = .02),
    sliderInput("T_h_rm", label = "Predator handling time",
                min = 0.001, max = 1.0, value = 0.3),
    sliderInput("e_rm", label = "Predator conversion efficiency", 
                min = .001, max = 1.0, value = 0.6),
    sliderInput("d_rm", label = "Per capita death rate of Predator",
                min = .0001, max = 1.0, value = .4),



    ### Ask users for initial conditions -----
    #N1, N2
    numericInput("H_rm", label = "Initial population size of Prey",
                 min = 1, value = 30),
    numericInput("P_rm", label = "Initial population size of Predator", 
                 min = 1, value = 25),

    ### Ask users for time to simulate ----
    numericInput("t_rm", label = "Timesteps", min = 1, value = 100),

    checkboxGroupInput("vectors_rm", label = "Display vector field?",
                       choices = c("Yes" = "Yes"), selected = "Yes")

  ),

  mainPanel(renderPlot(plot_rm()),
            renderPlot(np_rm())
            )

)

# Set the initial population sizes
init_rm <- reactive({c(H = input$H_rm , P = input$P_rm)})

# Set the parameter values
pars_rm <- reactive({
  c(r = input$r_rm, K = input$K_rm, 
    a = input$a_rm, T_h = input$T_h_rm, 
    e = input$e_rm, d = input$d_rm )})

# Time over which to simulate model dynamics
time_rm <- reactive({seq(0,input$t_rm,by = .1)})

out_rm <- reactive({
    data.frame(run_predprey_model(
    time = time_rm(),
    init = init_rm(),
    params = pars_rm()
  ))
  })

# Plots ------
## make abundance thru time plot
plot_rm <- reactive({
    plot_predprey_time(out_rm())
})

np_rm <- reactive({
  if("Yes" %in% input$vectors_rm) {
    plot_predprey_portrait(out_rm(), pars_rm(), 
                                         vectors_field = TRUE)
  } else {
    plot_predprey_portrait(out_rm(), pars_rm(), 
                             vectors_field = FALSE)
  }
})

Tuketici (Avcı) fonksiyonel yanıtı {.tabset}

Tüketicinin fonksiyonel yanıtı, kaynak tüketim oranını kaynağın yoğunluğunun bir fonksiyonu halinde belirtir. Başka bir deyişle, bu oran tüketicilerin kaynaktan enerji elde etme hızıdır ve kaynağın yoğunluğuna bağlıdır.

type1_fr <- function (H_init, H_final, a){
  to_return <- c(a*(H_init:H_final))
  return(to_return)
}

type2_fr <- function (H_init, H_final, a, Th){
  to_return <- c((a*(H_init:H_final)/(1+a*Th*(H_init:H_final))))
  return(to_return)
}

Tip I

Tüketim oranının kaynak yoğunluğuyla doğrusal artışı:

$$f(H) = aH$$

pars_vars <- c("$H$", 
               "$a$")
descriptions <- c("Population size of the prey",
                 "Attack rate of the predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    # a
    sliderInput("a_fr_type1", label = "Predator attack rate", min = .001, 
                max = 1.0, value = .02),

    ### Ask users for initial and final conditions -----
    #H_init, H_final

    numericInput("H_init_fr_type1", label = "Initial population size of Prey", 
                 min = 1, value = 30),
    numericInput("H_final_fr_type1", label = "Final population size of Prey", 
                 min = 1, value = 100), #Add warning: H_final>H_init 

  ),  

 mainPanel(renderPlot(plot_fr_type1())),

)


# Generate function output and data frame

out_fr_type1 <- reactive({type1_fr(H_init = input$H_init_fr_type1, 
                                   H_final = input$H_final_fr_type1,
                                   a = input$a_fr_type1)})

type1_fr_df <- reactive({data.frame(H = input$H_init_fr_type1:input$H_final_fr_type1,
                                    f_rate = out_fr_type1())})

#Plot

plot_fr_type1 <- reactive ({ggplot(type1_fr_df()) +
                            geom_line(aes(x = H, y=f_rate), size = 2) +
                            scale_color_brewer(palette = "Set1") +
                            xlab("Number of Prey") + 
                            ylab("Foraging Rate") +
                            ylim(0,50)})

Tip II

Doğrusal düşen tüketim oranı, tüketicinin doyum noktasında platolanır:

$$g(H) = \frac {aH}{1+aT_hH}$$

pars_vars <- c("$H$", 
               "$a$", 
               "$T_h$")
descriptions <- c("Population size of the prey",
                 "Attack rate of the predator",
                 "Handling time of the predator")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("Parametre/Değişkenler", "Tanımlar")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

sidebarLayout(
  sidebarPanel(
    ### Ask users for parameter values ----
    # a, Th

    sliderInput("a_fr_type2", label = "Predator attack rate", min = .001, 
                max = 1.0, value = .02),
    sliderInput("T_h_fr_type2", label = "Predator handling time", min = 0.001, 
                max = 1.0, value = 0.3),

    ### Ask users for initial and final conditions -----
    #H_init, H_final

    numericInput("H_init_fr_type2", label = "Initial population size of Prey", 
                 min = 1, value = 30),
    numericInput("H_final_fr_type2", label = "Final population size of Prey", 
                 min = 1, value = 100), #Add warning: H_final>H_init 

  ),  

 mainPanel(renderPlot(plot_fr_type2())),

)


# Generate function output and data frame

out_fr_type2 <- reactive({type2_fr(H_init = input$H_init_fr_type2, 
                                   H_final = input$H_final_fr_type2,
                                   a = input$a_fr_type2,
                                   Th = input$T_h_fr_type2)})

type2_fr_df <- reactive({data.frame(H = input$H_init_fr_type2:input$H_final_fr_type2,
                                    f_rate = out_fr_type2())})

#Plot

plot_fr_type2 <- reactive ({ggplot(type2_fr_df()) +
                            geom_line(aes(x = H, y=f_rate), size = 2) +
                            scale_color_brewer(palette = "Set1") +
                            xlab("Number of Prey") + 
                            ylab("Foraging Rate") +
                            ylim(0,4)})

Referanslar

Murdoch, Briggs and Nisbet tarafından Consumer Resource Dynamics - dijital kopyasına bir kütüphaneden ulaşılabilir.
M. L. Rosenzweig and R. H. MacArthur tarafından (1963) Graphical Representation and Stability Conditions of Predator-Prey Interactions.
Fonksiyonel ve numerik yanıt üzerine bir başlangıç
C.S. Holling tarafından (1995) Some characteristics of simple types of predation and parasitism.
C.J. Krebs et al. tarafından (1995) Impact of Food and Predation on the Snowshoe Hare Cycle.
A.N.P. Stevens tarafından (2010) Dynamics of Predation.

Konak ya da konakçi-parazit (ya da parazitoid) etkileşimi üzerine:
- Overivew of the Nicholson-Bailey model (Wikipedia linki)
- N.J. Mills and W.M. Getz tarafından (1996) Modelling the biological control of insect pests: a review of host-parasitoid models.

suppressWarnings(ecoevoapps::print_app_footer(language = "tr"))

gauravsk/ecoevoapps documentation built on July 9, 2024, 9:37 p.m.

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gauravsk/ecoevoapps
Simulate Dynamics of Ecology/Evolution Models

In gauravsk/ecoevoapps: Simulate Dynamics of Ecology/Evolution Models

{.tabset}

Lotka-Volterra Avcı-Av modeli

Lojistik Av

Tip 2 Fonksiyonel Yanit

Rosenzweig-MacArthur Modeli

Tuketici (Avcı) fonksiyonel yanıtı {.tabset}

Tip I

Tip II

Referanslar

R Package Documentation

Browse R Packages

We want your feedback!

gauravsk/ecoevoapps Simulate Dynamics of Ecology/Evolution Models

In gauravsk/ecoevoapps: Simulate Dynamics of Ecology/Evolution Models

{.tabset}

Lotka-Volterra Avcı-Av modeli

Lojistik Av

Tip 2 Fonksiyonel Yanit

Rosenzweig-MacArthur Modeli

Tuketici (Avcı) fonksiyonel yanıtı {.tabset}

Tip I

Tip II

Referanslar

R Package Documentation

Browse R Packages

We want your feedback!

gauravsk/ecoevoapps
Simulate Dynamics of Ecology/Evolution Models