​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Ecology Biology Chaos Education Population
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
JCU Environmental Biology
​Modelo retirado do link  https://insightmaker.com/insight/71649/Fern-Population-Model
Ecology Population Ecosystem Biology
​Modelo retirado do link 
https://insightmaker.com/insight/71649/Fern-Population-Model
Clone of Modelo da populacao de samambaias
joD. Roger
Un modello per l'effetto della temperatura (costante) sulla crescita di un pericoloso patogeno, agente di tossinfezioni alimentari (Listeria monocytogenes) __ Il modello è basato su questo Insight https://insightmaker.com/insight/206861/D-model-curve-di-Richards-con-ln-alpha-lag-mu
Richards Curves Biology Bacterial Growth Predictive Microbiology Microbiology
Un modello per l'effetto della temperatura (costante) sulla crescita di un pericoloso patogeno, agente di tossinfezioni alimentari (Listeria monocytogenes)

__
Il modello è basato su questo Insight https://insightmaker.com/insight/206861/D-model-curve-di-Richards-con-ln-alpha-lag-mu
Clone of Effetto della temperatura sulla crescita di L. monocytogenes
Anita Aliano
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Chaos Biology Population Ecology Education
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Hans Niedderer
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Chaos Education Ecology Biology Population
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Jay MJJ Gatsby
Un modello per l'effetto della temperatura (costante) sulla crescita di un pericoloso patogeno, agente di tossinfezioni alimentari (Listeria monocytogenes) __ Il modello è basato su questo Insight https://insightmaker.com/insight/206861/D-model-curve-di-Richards-con-ln-alpha-lag-mu
Microbiology Bacterial Growth Predictive Microbiology Richards Curves Biology
Un modello per l'effetto della temperatura (costante) sulla crescita di un pericoloso patogeno, agente di tossinfezioni alimentari (Listeria monocytogenes)

__
Il modello è basato su questo Insight https://insightmaker.com/insight/206861/D-model-curve-di-Richards-con-ln-alpha-lag-mu
Clone of Clone of Clone of Effetto della temperatura sulla crescita di L. monocytogenes
Maria Grazia
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Education Chaos Ecology Biology Population
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Nick May
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Education Chaos Ecology Biology Population
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Eleazar Puente
Simulation of MTBF with controls F(t) = 1 - e ^ -λt  Where   • F(t) is the probability of failure   • λ is the failure rate in 1/time unit (1/h, for example)  • t is the observed service life (h, for example) The inverse curve is the trust time On the right the increase in failures brings its
Navier Stokes Health Physics Population Risk MTBF Failure TURBULENCE Biology Growth BIFURCATIONS Environment Economics Finance Chaos Mathematics Demographics Engineering Fractals Strategy Science
Simulation of MTBF with controls

F(t) = 1 - e ^ -λt 
Where  
• F(t) is the probability of failure  
• λ is the failure rate in 1/time unit (1/h, for example) 
• t is the observed service life (h, for example)

The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime, 
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life
If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

Useful Life
The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.  

Wearout
The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period. 
Clone of Clone of Clone of BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK
Guilherme Werpel Fernandes
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
Ecology Environmental Science Population Dynamics Biology
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
Clone of Plant, Deer and Wolf Population Dynamics G-IV Intro
Leonard Kosub
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Chaos Population Biology Ecology Education
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Richard Jenkins
This simulation shows how algae, tadpole and dragonfly populations impact each other in a pond ecosystem.
Population Dynamics Environmental Science Biology Ecology
This simulation shows how algae, tadpole and dragonfly populations impact each other in a pond ecosystem.
Algae, Tadpole and Dragonfly Population Dynamics
Laryssa Faith Laurignano
Insight diagram
Population Ecology Biology Ecosystem
Clone of Modelo da populacao de samambaias
Beatriz Neri Nascimento
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Ecology Biology Population Chaos Education
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
reno marcello m
Two coupled Lotka-Volterra models, in one of which there are two predators
Biology Dynamical Systems Environment
Two coupled Lotka-Volterra models, in one of which there are two predators
Clone of 4 species
Tommaso Corridoni
Insight diagram
Biology Food Chain FOOD WEB
Clone of Food Chain
joe dowling
Insight diagram
Food Chain FOOD WEB Biology
Clone of Food Chain
Owen Walker
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Education Chaos Population Biology Ecology
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
christian hitti
Insight diagram
Biology
wolf and moose
vivianne
Insight diagram
Biology
Moose-Wolf
Jetaime Lampinen
Simulation of MTBF with controls F(t) = 1 - e ^ -λt  Where   • F(t) is the probability of failure   • λ is the failure rate in 1/time unit (1/h, for example)  • t is the observed service life (h, for example) The inverse curve is the trust time On the right the increase in failures brings its
TURBULENCE Demographics Fractals Physics Science Failure BIFURCATIONS Strategy Growth Risk Economics Navier Stokes Finance Engineering Biology Chaos Mathematics Environment Health Population MTBF
Simulation of MTBF with controls

F(t) = 1 - e ^ -λt 
Where  
• F(t) is the probability of failure  
• λ is the failure rate in 1/time unit (1/h, for example) 
• t is the observed service life (h, for example)

The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime, 
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life
If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

Useful Life
The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.  

Wearout
The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period. 
Clone of BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK
Nilo Guimaraes
Prey dx / dt =  αx  -  βxy The prey reproduces exponentially ( αx ) unless subject to predation. The rate of predation is the chance ( βxy) with which the predators meet and kill the prey. Predator dy/dt = δxy - γy The predator population growth δxy depends on successful
Biology Population Education Ecology Chaos
Prey
dx/dt = αx - βxy
The prey reproduces exponentially (αx) unless subject to predation. The rate of predation is the chance (βxy) with which the predators meet and kill the prey.

Predator

dy/dt = δxy - γy

The predator population growth δxy depends on successful kills and the reproduction rate; however, delta is likely be different from beta. The loss rate, an exponential decay, of the predators {\displaystyle \displaystyle \gamma y}γy represents either natural death or emigration

Ex2b: prey vs predator model
Rolf Widmer
​Physical meaning of the equations The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations: 1. The prey population finds ample food at all times. 2. The food supply of the predator population depends entirely on the
Ecology Education Chaos Biology Population
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dt = αx - βxy
 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.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} 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). {\displaystyle \displaystyle \gamma y} 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.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Clone of Prey&Predator
Yezabel Rincón
Insight diagram
Food Chain FOOD WEB Biology
Clone of Food Chain
Krirushan