---
title: "A critique of the Lotka Volterra predator prey model"
author: "Duncan Golicher"

output:
  slidy_presentation: default
  ioslides_presentation: default
  beamer_presentation: default
  powerpoint_presentation:
bibliography: [ecosystems1.bib, ecosystems2.bib,insects.bib]
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE, warning = FALSE,message = FALSE)

```

# The Lotka Volterra predator prey model

$\frac{dN}{dt} =r_{prey}N-\alpha NP$

$\frac{dP}{dt} =\alpha \beta NP- m_{predator}P$

Where N = number of prey, $r_{prey}$ = intrinsic growth rate of prey, $\alpha$ = proportion of encounters between prey and predator leading to mortality of prey, P= numbers of predators $\beta$ = conversion of prey to predator and $m_{predator}$ = predator mortality rate

<https://insightmaker.com/insight/4DYFdNOfy5alJd395LWc12/Lotka-volterra>

# Lotka Volterra model

![](figs/lvmp.png){width="400"}

# What is assumed in the classic model?

-   The model assumes complete "top down" regulation of the prey population (no carrying capacity for prey in absence of predation)
-   Prey mortality that prevents exponential growth is only due to predation.
-   So, in the absence of predation, the model leads to exponential growth of prey
-   Likewise, the predator population is only regulated by the availability of one sort of prey (no alternative food)
-   The model assumes that predators gain energy only through prey consumption
-   Hunting efficiency and conversion efficiency (prey to predator energy transfer) are fixed.
-   Predator behaviour not considered.
-   Prey age structure ignored.

# Is this model useful?

-   It can be argued that the model is useful as a thought tool for considering what might happen within such a simple system
-   It has been argued that some high latitude (arctic and sub arctic) ecosystems may approximate the conditions in the model.
-   Classic example: Lynx and snowshoe hares in Canada.
-   Arctic foxes and lemmings

# Is the evidence compelling?

-   The snowshoe hare/Lynx example has been extensively analysed.
-   Gilpin suggested that the cycles in the data don't match the model[@gilpin1973]
-   The behaviour of the trappers and the market for pelts may explain the cycles.[@deng2018]

![](figs/hare.png){width="300"}

# Are prey regulated by "bottom up" factors?

-   Prey numbers must depend on the availability of food
-   Adding a "carrying capacity" to the model "damps down" any cycles
-   Food availability varies over time both within years (seasonality) and between years
-   Some apparent cycles may be due to rapid increases in prey in "good years"

![](figs/lemming.png){width="200"}

# What happens in more complex systems?

-   Most predators feed on a range of different prey species
-   Consumption may be spread over prey that feed on different foods. This will reduce top down regulation.
-   Predators may switch attention towards the most abundant prey species
-   Not all predators are strictly carnivorous (bears for example, or urban foxes)
-   Scavenging may also add to food supply

![](figs/fox.png){width="200"}

# Which elements in the prey population are most exposed to predation?

-   Predators will tend to take prey that are easy to catch.
-   Young (weak or inexperienced)
-   Old and/or sick individuals
-   These may be vulnerable to other forms of mortality, thus reducing the impact of predation on the population dynamics.

# Conditions that weaken prey may be good for predators

-   Harsh winters or droughts will reduce the birth rate of prey, but may help predators catch them.

-   This may produce dynamics that directly contradict the model

![](figs/harsh.png){width="200"}

# Evolutionary "arms races"

-   Selection pressure acts on prey to enhance their ability to escape predation
-   Selection pressure acts on predators to enhance ability to catch prey
-   "Fit" i.e. mature healthy prey may be very difficult to catch

![](figs/cheetah.jpg){width="300"}

# Predators vs scavengers

-   The distinction between predation and scavenging may not be clear cut
-   Many carnivores play both roles
-   Pure scavenging has no impact on the prey population

![](figs/hyaenas.png){width="300"}

# Other forms of predation

-   Some "predators" specialise on specific age groups

-   For example, nest predation of birds

-   Understanding the impact of this requires consideration of an age structured population

![](figs/magpie.png){width="200"}

# Functional response

-   The Lotka Volterra model does not consider the dynamics of predation under varying prey densities.

-   When prey density is low, more time must be spent searching for prey

-   When prey density is high, the rate of consumption may saturate (handling time reached)

![](figs/waders.png){width="220"}

# Functional response

```{r, fig.width=8, fig.height=4}
library(tidyverse)
Density<-1:100
f<-function(x,s=2,F=30)x*s/(F+x)
d<-data.frame(Density,Consumption=f(x=Density))
ggplot(d,aes(x=Density,y=Consumption))+ geom_line() +labs(title="Functional response",x="Prey density",y="Numbers of prey consumed per predator") +  theme(axis.text.x = element_blank(),axis.ticks.x = element_blank(),axis.text.y = element_blank(),axis.ticks.y = element_blank()) + theme_bw()
```

# What does coexistence of predators and prey imply?

-   For predators coexist with prey the system is likely to be more complex than the simple model

-   All predator prey systems do seem to be more complex in some respect than the model

-   It is possible that predation has led to extinctions in the past (difficult to know)

-   Predators are likely to regulate prey populations to some extent, but it may be difficult to predict how much top down regulation occurs.

# Humans as predators

![](figs/hunter.png)

# Impact of hunting

-   Prey have not generally had time to evolve sufficiently to hunting

-   Hunting can be indiscriminate (takes healthy, mature individuals of reproductive age)

-   Human "predation" may have very different effects on prey populations than natural predation.

-   Hunting can be regulated or deliberately designed for management purposes

-   Hunting that mimics natural predation could be preferable when it is impossible to reintroduce predators

# So, where does this leave the Lotka Volterra model?

-   Very limited evidence that cycles are due to the mechanism proposed in the model

-   Assumptions too simplistic

    -   Only top down control

    -   No prey switching or alternative diets

    -   No consideration of predator behaviour

    -   No consideration of age structure of the prey

    -   Simplistic depiction of predation

-   But, it produces interesting dynamics!

# What does this imply regarding models in general?

-   Models are useful when assumptions are tested.

-   Models help to provide a formal framework for thought

-   Mathematical tractability and analytical sophistication (hard maths to follow) do not make models more "scientific"

-   Interesting model behaviour should not be a criterion for further research into the model aimed at understanding the system it depicts.

# Are other models helpful?

-   All models, whether their assumptions and predictions are supported or not, do help to provoke constructive thought.

-   Some models provide a very solid framework for thought. E.g. the logistic equation helps to formalise the concept of the carrying capacity, which is a very reasonable assumption

-   Some models are extremely useful for short term projections (e.g the Leslie matrix)

-   Some models help to formalise general hypotheses (e.g. the intermediate disturbance hypothesis)

-   Some complex models can help in making predictions regarding even more complex systems, providing they are always open to criticism and analysis.

# References
