Ph. D. Scholar Department of Entomology Indira Gandhi Krishi Vishwavidyalaya Raipur (Chhattisgarh)

1. PREDATOR-PREY MODELS-
LOTKA-VOLTERRA,NICHOLSON–BAILEY
MODEL , CROP MODELLING
Prepared By
Aaliya Afroz
Ph. D. Scholar
Dept. of Entomology, IGKV, Raipur

2. Introduction
• These models in regards their relation to classical biological control,
which is the practice of introducing a natural enemy of an insect pest with
the goal of reducing the insect pest's population to a level that is no longer
dangerous to the local region.
• Predator-prey models are
arguably the building blocks of
the bio and ecosystems as
biomasses are grown out of their
resource masses.

3. Basic Framework of Models
• Models have been favoured due to the following three assumptions
that can be made:
1. Closed system:
2. Equivalent generation times
3. Ignore/simplify age structure

4. ØThere are two major categories for host-parasitoid
models which depends on how generations are
measured in time.
These two categories are
Discrete generations
Continuous
generations

5. TYPES OF MODEL
Basically models are categorised in two broad categories-
• Early models
• Refinements of early models

6. Early models Refinements of the Models
vLotka-Volterra Model
vThompson Model
vNicholson-Bailey Model
vHolling Model
ØDensity-Dependent Self-Limitation
ØTemporal Limitations:Age-Structure

7. Lotka-Volterra Model
What is lotka volterra model?
• Lotka volterra model provide information about
competion between two species living in same
ecosystem.
• Lotka volterra model help us predicting the outcome
of a competition based on impact of one species on
another.

8. History
• Initially proposed by Alfred J. Lotka in
the theory of autocatalytic chemical
reactions in 1910.
• In 1925 he used the equations to analyse
predator– prey interactions in his book
on biomathematics.
• The same set of equations was published
in 1926 by Vito Volterra, a
mathematician and physicist, who had
become interested in mathematical
biology.
Alfred J. Lotka
Vito Volterra

9. Assumptions of lotka- velterra
equation
• Animals move at random.
• Every encounter with a prey results in a capture and every prey that
is captured is eaten. This is independent of densities of population of
predation and prey.
• The population of both predator and prey have all the qualities
necessary for them to conform logistic theory –growth rate
accelerates until alimiting factor causes deceleration.

10. Lotka–Volterra equations
• The Lotka–V
olterra equations, also known as the predator–prey
equations, are a pair of first order nonlinear differential equations.
• Frequently used to describe the dynamics of biological systems in
which two species interact, one as a predator and the other as prey.

11. Prey equation
• 𝜹𝒙/ 𝜹𝒕 = 𝜶𝒙 – 𝜷𝒙
• 𝒙 is the number ofprey
• 𝒚 is the number of somepredator
• 𝜹𝒙/ 𝜹𝒕 and 𝜹𝒚/ 𝜹𝒕 represent the instantaneous growth rates of the two
populations.
• t represents time
• α, β, γ, δ are positive real parameters describing the interaction of the two
species.

12. About prey equation
• The prey are assumed to have an unlimited food supply and to
reproduce exponentially, unless subject to predation.
• This exponential growth is represented in the equation above by the
term 𝜶𝒙.
• The rate of predation upon the prey is assumed to be proportional to
the rate at which the predators and the prey meet, this is represented
above by 𝜷𝒙𝒚.
• If either 𝒙 or 𝒚 is zero, then there can be no predation.

13. P r e d a t o r equation
• 𝜹𝒚 /𝜹𝒕 = 𝜹𝒙𝒚 – 𝜸𝒚
• • 𝒙 is the number ofprey
• • 𝒚 is the number of somepredator
• • 𝜹𝒙/ 𝜹𝒕 and 𝜹𝒚/ 𝜹𝒕 represent the instantaneous growth rates of the two
populations.
• t represents time;
• α, β, γ, δ are positive real parameters describing the interaction of the two
species.

14. About p r e d a t o r equation
• In this equation, 𝜹𝒙𝒚 represents the growth of the predator population.
• Note the similarity to the predation rate; however, a different constant is
used, as the rate at which the predator population grows is not
necessarily equal to the rate at which it consumes the prey.
• 𝜸𝒚 represents the loss rate of the predators due to either natural death
or
emigration, it leads to an exponential decay in the absence of prey.

15. Predator prey model’s assumptions
1.The prey population finds ample food at all times.
2.In the absence of a predator, the prey grows at a rate
proportional to the current population.
3.The food supply of the predator population depends entirely
on the prey populations.
4. In the absence of the prey, the predator dies out.
5.The number of encounters between predator and prey is
proportional to the product of their populations. Encounters
between predator and prey tends to promote the growth of the
predator and inhibit the growth of the prey.
6.During the process, the environment does not change in favor
of one species and the genetic adaptation is sufficiently slow.

16. Nicholson and Bailey model
• This model is also known as host-parasitoid model.
• The Nicholson–Bailey model was developed in the 1930s to describe
the population dynamics of a coupled host-parasitoid system.
• It is named after Alexander John Nicholson and Victor Albert Bailey.
• Host-parasite and prey-predator systems can also be represented with the
Nicholson-Bailey model.
• The model is closely related to the Lotka–Volterra model, which
describes the dynamics of antagonistic populations (preys and predators)
using differential equations.

17. Alexander John Nicholson Victor Albert Bailey
History
The classic model was discribed by Nicholson and Bailey in
1935.

18. Derivation
• The model is defined indiscrete time.It is usually expressed as-
Nt+1 = λNt e-aP
t
Pt+1 = cNt[1 -e-aP
t]
Where,
Nt = density of host species in generationt
Pt = density of parasitoid in generation t
a = searching efficiency of parasitoids
λ = host reproductive rate
c = average number of viable eggs laid by a parasitoid on a single host.

19. Case Studies
Winter moth
• In the 1950s, winter moths were harming
hardwood trees in eastern Canada, so a parasitoid
was introduced from the moths' native Europe,
which greatly reduced the abundance of winter
moths. This reduction was most easily modeled at
the time with the Nicholson-Bailey model and
aggregrated parasitoid attack rates, but upon
review of the data, the winter moth case study
appeared to be more a matter of predation that
parasitism because of the decline in unparasitized
pupae in the soil.

20. Cassava mealy bug & California red
scale
• More recent application of host-parasitoid interactions include the cassava
mealybug, an insect pest found in Africa, and the California red scale, which
is found in California.
• The interactions of these populations with their respective parasitoid
models, and some
populations were modeled using Lotka-Volterra
interesting conclusions were reached for both systems:
• Very little evidence of aggregrated parasitoid attack.

21. • Age structure, in some form, is
essential in order to correctly model
host-parasitoid interactions.
• Local dynamics of the interactions
are strongly influenced by parasitism
refuges, in the form of physical
refuges or host quality effects.
• Temporal refuges (i.e. syncroization
issues) were not apparent, and host
feeding did not significantly change
overall dynamics.

22. A TWO-PATCH PREY-PREDATOR MODEL WITH
PREDATOR DISPERSAL DRIVEN BY THE PREDATION
STRENGTH
• Foraging movements of predator play an important role in population dynamics
of prey-predator systems, which have been considered as mechanisms that
contribute to spatial self-organization of prey and predator.
• In nature, there are many examples of prey-predator interactions where prey is
immobile while predator disperses between patches non-randomly through
different factors such as stimuli following the encounter of a prey.
• In the work, formulated a prey-predator two patch model with mobility only in
predator and the assumption that predators move towards patches with more
concentrated prey-predator interactions.
• It provide completed local and global analysis of model.

23. • Analytical results combined with bifurcation diagrams suggest that:
ü Dispersal may stabilize or destabilize the coupled system
ü Dispersal may generate multiple interior equilibria that lead to rich
bistable dynamics or may destroy interior equilibria that lead to the
extinction of predator in one patch or both patches
ü Under certain conditions, the large dispersal can promote the permanence
of the system. In addition, we compare the dynamics of our model to the
classic two patch model to obtain a better understanding how different
dispersal strategies may have different impacts on the dynamics and
spatial patterns.

24. Epidemics in predator–prey models:
disease in the predators
• The investigated models for the study of interacting species subject to an
additional factor, a disease spreading among one of them, that somehow
affects the other one. The inadequacy of such a model comes from the
basic assumption on the interacting species. It is well known that the
cycles found in the Lotka–Volterra system exhibit a neutral stability, and
this phenomenon is carried over to the proposed model. Here we would
like to extend the study to account for population dynamics leading to
carrying capacities, i.e. logistic behaviour.
• This corresponds to the so-called quadratic predator–prey models found in
the literature.

25. Mean free-path length theory of
predator–prey interactions: Application
to juvenile salmon migration
• Ecological theory traditionally describes predator–prey interactions in terms
of a law of mass action in which the prey mortality rate depends on the
density of predators and prey.
• This simplifying assumption makes population-based models more tractable
but ignores potentially important behaviors that characterize predator–prey
dynamics.
• Here this model expand traditional predator–prey models by incorporating
directed and random movements of both predators and prey. The model is
based on theory originally developed to predict collision rates of molecules.

26. Importance of Crop Modeling in
Agriculture with reference to Pest
Management

27. An agricultural system, or agro-ecosystem, is a collection of
components that has as its overall purpose the production of
crops and raising livestock to produce food, fiber, and energy
from the Earth's natural resources and such systems may
also cause undesired effects on the environment. (Jones et al.,
2016).
Introduction

28. What is a model?
ØA physical, mathematical, or otherwise logical
representation of a system, entity, phenomenon,
or process (DoD 1998).
ØA representation of one or more concepts that may be
realized in the physical world (Friedenthal, Moore, and
Steiner 2009).
ØA simplified representation of a system at some particular
point in time or space intended to promote understanding of
the real system (Bellinger 2004).

29. vMathematical Model - Physical relationship of natural phenomenon by
Means of a mathematical equation are called mathematical Model .
vGrowth Model - If the phenomenon is expressed in the growth define it
is define as growth model
vCrop Weather Model - Crop weather model is based on the principle
that govern the development of crop and its growing period based on
temperature and day length .
Types of Models

30. Ø Crop modeling helps in yield predicting and forecasting
Ø Helps in evaluation of weather change, thus helps in weather forecasting
Ø Helps in formation of stocks, making of agricultural policies and zoning
Ø Optimum seed rate can be calibrated from these models
Ø Useful in cropping management system by predicting cultural practices
Ø Helps to quantify optimum amount of fertilizer and decide optimum time
of application
Ø Helps to predict pests outbreak through crop weather model
Application of Crop Modeling in Agriculture

31. Some Crop Models Reported in Recent Literature
Software Details
SLAM II Forage harvesting operation
SPICE Whole plant water flow
IRRIGATE Irrigation scheduling model
COTTAM Cotton
CropSyst Wheat & other crops
TUBERPRO Potato & disease
WOFOST Wheat & maize, Water and nutrient
WAVE Water and agrochemicals
ORYZA1 Rice, water
SIMCOY Corn
APSIM-Sugarcane
Sugarcane, potential growth, water and
nitrogen stress

32. Model uses
Simulation modelling is increasingly being applied in
research, teaching, farm and resource management, policy
analysis and production forecasts.
These model can be applied into three areas, namely;
ØResearch tools,
ØCrop system management tools, and
ØPolicy analysis tools.

33. As research tools
Ø Research understanding
Ø Integration of knowledge across disciplines
Ø Improvement in experiment documentation and
data organization
Ø Genetic improvement
Ø Yield analysis

34. As crop system management tools
Ø Cultural and input management,
Ø Risks assessment and investment support
Ø Site-specific farming

35. As Policy Analysis Tools
Ø Best management practices
Ø Yield forecasting
Ø Introduction of a new crop
Ø Global climate change and crop production

36. Agricultural systems are characterized by high levels of interaction
between the components that are not completely understood.
Lack of knowledge and data can give rise to simplified
representation of a rather intensive system.
The need for model verification in a new situation arises because
all processes are not fully understood and even the best
mechanistic model still contains some empiricism making
parameter adjustments vital in a new situation.
Model performance is limited to the quality of input data.
Limitations: