The Lotka Volterra predator prey model

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

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

• N = number of prey

• P= numbers of predators

• \(r_{prey}\) = intrinsic growth rate of prey

• \(\alpha\) = proportion of encounters between prey and predator leading to mortality of prey

• \(\beta\) = conversion of prey to predator and

• \(m_{predator}\) = predator mortality rate

Lotka Volterra model

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

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(Gilpin 1973)
• The behaviour of the trappers and the market for pelts may explain the cycles.(Deng 2018)

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”

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

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

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

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

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

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)

Functional response

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

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.