REM 611

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Define: Ecology: Study of Ecosystems
Population: Group of individuals of one species in one area
Community: Species which occur together in space and time
Why are is it important to define your definitions of Ecology, Population and Community?: The often defined very arbitrarily and it is essential to be clear in your study so that it can be applicable for another study.
What is a model?: An abstraction of reality.
Includes componants of interest
Relationships of interest
There is no single best model because situations changes
What did Einstein say about simplicity?: "Make everything as simple as possible, not simpler."
What are the two types of models?: Mathmatical-Computer
Physical-Pictoral
Physical Model: Building scale models to help interpret and promote the success of a project. (Example with blue printers and modelers in the floor below)
Pictoral Model: Incredibly powerful tool for communication

Visual expression of model

Makes elements of model very clear
Purpose of a model: Describe
Communicate
Predict
Make Decisions
Understand
Defend Decisions
How are models used in defending a decision?: The analyst is able to document their rational and thinking process which led them to the decision and if it doesn't work out then their is guidence for the next trial.
How are models a good substitute for experiments?: 1) If the species is endangered and the risk is too high it is better to simulate the experiement where possible.

2) Things of high monetary value are difficult to convince people to change for the sake of an experiment. It is better to simulate in this case.
How do models help formulate hypotheses?: Models can be used to test hypotheses and predict results.
Sensitivity Analysis: Identifying Key Componants of a Model

Parameters in a model which do not have much leverage on accuracy are not worth the money and time to put in the model.
Analytical Model: Able to solve directly for an answer
Simulation Model: Solved Repetedly to get an answer
Dynamic Model: t shows up as a subscript - there is an explicit time componant
Static Model: E=MC squared
Empiracle Model: Summarizes relationships with no regard for the mechanism

- a stastical model particularly a linear regression model is an example of this

-it describes the data, not the rationale
Mechanistic Model: Explicity includes mechanism
Stochastic Model: Includes parameters for randomness
Deterministic Model: Single predicted outcome

Repeatable results
Non Spatial Model: No parameters for space

Tree model with just Nt
Spatially Explicit Model: Precise location

Tree example with Nt and x and y
Spatially Implicit Model: Doesn't specify specific location

Only provides information about parameters relative to each other
Discrete Model: Time passes in "chunks"

Precces occur as events at distinct times followed by intervals when notheing happens
Coninuous Model: The change in t is infentessimally small

Processes occur continuously
Discrete Exponential Growth Model Formula: Also known as the Difference Model

Difference between births of one year to the next

Nt+1=Nt+Ntb

Where the population next year equals the population now plus the number of babies born (product of # of individuals times birth rates)
Continuous Model Formula: dN/dt=rN where the change in N with a very small has some % impact on N
Intrinsic Rate of Increase: The ability of the species to concert that extra energy (aquired by extra food) into enhaced fecundity and diminished mortality
Fecundity Rate - Explaination and Formula: Birth Rate

Number of female babies born per female per time (usually 1 year)
Mortality Rate: Number of animals which die at time t
IFD Conclusions: Prediction 1 doesnâ€™t hold true, while prediction 2 does
Habitat Matching Rule Formula: If S1=S2
Then E1/N1=E2/N2
Therefore N2/N1=E2/E1

Where N is population and E is energy intake
Habitat matching rule definition: It is a surrogate method for determining fitness for an organism based on habitat use

The habitat with more â€œfishâ€ will have more food
IFD Prediction 2: â€œHabitat Matching Ruleâ€
IFD Prediction 1: When all habitats are occupied the mean suitability across all habitats should be identical
Basic Suitability: Fitness level before any other organism enters area
IFD Model Parameters: Si=Bi-Fi(Di)

Suitability for individuals is determined by basic suitability minus some function of density dependence AKA decreasing effect of density
â€œIdeal Assumptionâ€: Perfect Information
Assumptions of Fretwell and Lucas: Organism will select habitat
This is an evolved trait
Suitability of any habitat will decrease as a function of density
Allee Effects arenâ€™t included
Organism will only settle in most suitable habitat (Which gives highest fitness)
Assume al individuals achieve same fitness
Organism free to enter any habitat
If all assumptions are true than will see IFD
Habitat suitability: â€œgoodnessâ€ or contribution to gene pool of future generations
Distribution: # or proportion of individuals in each habitat
Example of habitat discrimination: H1: Low food, high predation
H2: High food, low predation
Habitat Assumptions: Habitats may vary in physical and biological features which contribute to survival

Organisms have the ability to differentiate between god an bad habitats
Habitat: Portion of earthâ€™s surface where and organism colonizes and lives
Three Hypotheses of IFD: 1) Territoriality may be a mechanism for distribution

2) Territoriality may be a modifier for distribution patterns â€“ an add on

3) Territoriality may be a way of preserving distribution but not a motivator for it
Home Range: An area where an organism spends most of its time
Territoriality: Conspecific behavior interactions dealing with conspicuousness
What question does IFD answer?: Why are organisms where they are at what time
What does IFD stand for?: Ideal Free Distribution
Who developed the IFD: Fretwell and Lucas 1970
Fitness: Maximizing survival and reproduction
What in general have we not considered for the models we have used thus far?: The organismâ€™s fitness level
What assumptions have been made in Schoener model 2?: Assumed energy intake was the same for all animals
What assumptions have been made in Schoener model 1?: We assumed all animals breed
What is important about the way organisms are distributed?: This info can inform us about their decision making process and aid in making predictions
Then why was the development of metapopulations â€œgoodâ€?: Helped inspire interest in dispersal
Helped people understand local extinctions better
Acknowledged it was difficult to assess habitat quality
What is the downside of metapopulations?: People can argue that extinction is natural
They also donâ€™t manage for biodiversity
Describe the source sink: One or a few â€œlargeâ€ patches feeding into many â€œsmallâ€patches where lambda in the large patch is >1 and in the small patches it is <1
Rescue Effect: Effect of overflowing source population recolonizing recently extinct metapopulation
Who developed the source sink model AKA the Core-satellite distribution?: Doak and Mills 1994
Who developed the idea of metapopulations?: Levins 1969
What is Levinâ€™s metapopulation equation?: dp/dt = m p (1 - p) - e p ,
where p is the proportion (fraction) of population centers (e.g., habitat "islands" or patches), m is the migration (colonization) rate, and e is the rate at which local populations go extinct.
At equilibrium p* = 1 - e/m. The metapopulation will persist (i.e., p* > 0) only if e < m.
Example of Fugitive Species: Fireweed
Fugitive Species: Species which have evolved to exploit fragmented niche (but not dominate â€“ invasive species
In the metapopulation model local Extinction Events are followed by: Re-colonization
Examples of Fragmentations: Roads, Agriculture, Forestry
Metapopulation: Ensemble of interacting populations which each have a finite life
Fragmentation: Previously continuous landscape divided into patterns surrounded by inhospitable matrix
What can we determine with Schoenerâ€™s mechanistic model?: N* or K
Limitations with Schoener Model 2 Graph: Doesnâ€™t make sense for range which has very few individuals in it because unlimited babies cannot occur
Schoener Model 1 (synonym): â€œTime Limitation Modelâ€
Common Reactivity Field in Africa: Watering Hole
Schoener Model 2 (synonym): "Energy Limitation Model"
Intraspecific Interference Competition: There is a cost associated with interacting with conspecifics
Logistic Continuous Formula: dN/dt=rN(1-N/K)
Why does the logistic model work? Schoener Reason 2: Logistic model is just a convenient â€œgood fitâ€ for lots of empirical data sets
Why does the logistic model work? Schoener Reason 1: It is a literal representation of the biological process resulting in population growth
Empirical Models: Summarizes relationships with no regard to mechanism
Schoenor 1973 sought to answer what question?: Why does the logistic model work?
Schoenor: â€œShanerâ€
Wide variety of models based on the logistic model occur when the following assumptions are manipulated:: Competition
Resource Acquisition
Time Allocation
Resource Availability
Problems with â€œRoad Kill Scienceâ€: Limited by type of death
(Easy sampling though)
Problems with following the entire cohort: Determining Mortality
Environmental Stochasticity
Length of time required to wait may be very long
Problems with Mark-Recapture Method: Determining Mortality
Difficult to judge age
How do you measure survivorship?: Mark-Recapture
Follow an entire cohort throughout their life cycle
â€œRoad Killâ€ Method
Graphical Lambda =: Plot LnN vs T
e^r (slope)
Graphical Method for simulating Lambda collapse: Determining stable age structure through simulation
Pick initial starting population values
Iterate until reach stable age structure
Two methods for Lambda collapse: Graphical
Analytical
If Lambda <1 If Lambda=1 If Lambda>1: Negative exponential growth
0 net growth
Exponential growth
Lambda: Discrete per capita rate of growth of a population
Age/stage class models generally model F or M?: Females although, males have a lower r if they are a trophy species
Assumptions of the Discrete Model: Time passes in chunks
Same lambda for all individuals
Lambda doesnâ€™t change with environment or density
No emigration or immigration
Age or Stage class models: Models used to describe populations where different age (stage) individuals have different probabilities of dying, giving birth etc.
Recommendations for avoiding the pitfalls of pseudorandomness: Look up seed
Use good seeds
Never trust RNG (look up)
Use multiple RNGs
Describe your RNG method
Seed: Method of instructing where RNG begins from
Pseudorandom: RNG always begins from the top/or same number in its list
Example of RNG: RANDBETWEEN function in excel
RNG: "Random Number Generator⬝
POO: Probability of Occurrence
Continuous Probability: Value can be any within a particular range
Binomial Probability: A and B Events
Example of Discrete Probability: 1 or 2 pine cones, or 2.2 and 3.2 (Nothing in between)
Discrete Probability: Events which can take only particular values
Two classes of probability: Discrete and Continuous
Significance of Independent Sample: Important to determine randomness so one can know whether or not to treat samples independently
Example of up and down random test: Stock market
Example of test for randomness: computer evalutes output for patterns
1-D Random Numbers 2-D Randomness 3-D Randomness: Time or Space
Time and Space (Raindrops on paper) x and y
X,Y,Z
â€œGamblers Fallacyâ€: Current events are not influenced by previous outcomes
Random Number: With out aim definition or direction
Example of Genetic Stochasticity: Fur color changes
Genetic Stochasticity: Random changes in gene frequencies (Not natural selection)
Example of Natural Catastrophes: Example of Natural Catastrophes Fire, Hurricane
Natural Catastrophes: Infrequent but devastating events experienced by all individuals
Example of Environmental Stochasticity: Summer weather variation
Environmental Stochasticity: Random events experienced similarly by all individuals (Frequent)
What is an example of demographic stochasticity?: Luck at finding food, parental skills, nest making materials etc.
Demographic Uncertainty: Random events experienced differently for different individuals
What did Shaffer proposed occurred in natural systems?: That there were four sources of uncertainty: Demographic, Environmental, Genetic and Natural Catastrophes
It is more realistic to have a range of predictions because_______: There is uncertainty in the real world
Even though there is geometric growth, ___ can still vary.: R
Deterministic: Always the same, every time we run a model it is the same prediction
what are two ways that models have been used in the management of wildlife populations?: a) The control of populations which are of socioeconomic concern
b) The conservation of endangered species
State in your own words the problem raised by Beissinger and Westphal (1998) with the way models are used.: The problem Beissinger and Westphal pointed out is that these PVA models are being used without any regard to the specifics of the population characteristics. Density dependence is not being taken into account among other things. The PVA models are also being used without first determining which type of model explains the observed patterns the best.
What is the equation for the discrete geometric growth model?: a. Nt+1=Nt+rNt
Write a new version of the discrete geometric growth model that includes the removal of y% of the rhinos each year: b. Nt+1=(Nt+rNt)-y%Nt
What is the equation for the discrete logistic growth model?: Nt+1=Nt+Ntr(1-Nt/K)
Write a new version of the discrete logistic growth model that includes the removal of y% of the rhinos each year.: Nt+1=Nt+Ntr(1-Nt/K)-y%Nt
is the equation for the discrete fowler growth model?: Nt+1=Nt+Ntr(1-(Nt/K)^n))
Write a new version of the discrete fowler growth model that includes the removal of y% of the rhinos each year.: Nt+1=Nt+Ntr(1-(Nt/K)^n))-y%Nt
State in your own words why the Fowler version of the model would be used instead of the logistic model.: The fowler version involves adding the exponent n to the growth rate parameter of the model to account for the heavier effect of density dependence among larger animals in the real world.
What are two things that could be done to decide which of these models was â€œbestâ€: A . Use correlation coefficients to evaluate which model has the best â€œfitâ€ and use for comparison.

B. Compare growth rate with body of scientific knowledge/historical observations to determine if the growth rates are biologically possible for this type of organism.
, we saw that we could use calculus to integrate this equation and derive the following analytical solution:: N(T) = N(0)erT
continuous geometric growth⬝ model:: dN/dt = rN
Fortunately, there are several⬝tricks⬝ for approximating the population trajectories using these differential equations. The one we will use is called: Eulerâ€™s Method
Eulerâ€™s Method: Nt+h = Nt + (dN/dt) * h
) What is the equation to describe the Euler approximation to the continuous geometric growth model: Nt+h = Nt + (rN) * h
So to get a better approximation, should h be larger or smaller? Why: A smaller h is better because it takes into account more infinitesimally small time increments which bring it closer to representing a continuous function, or its exact analytical counterpart.

in other words, the slope dN/dt gets calculated more frequently
In class, we talked about 8 reasons why population growth rate could be density dependent: ⬢ Aggression
⬢ Food Scarcity
⬢ Predators concentrate efforts more
⬢ Disease
⬢ Interference (Eating etc)
⬢ Limited living space
⬢ Predators learn better methods for killing prey
⬢ Immigration declines
⬢ Emigration increases
What is the analytical solution to the differential equation that describes continuous geometric growth: N(T) = N(0)erT
Give two sentences describing the potential problem with the North Sea cod stock as presented by the authors with respect to the current (as of 1997) fishing practices.: a. The current exploitation regime is unsustainable.
b. Even managing for the maximum sustained yield may be unsustaible because of the inherent risks in the system.
Based on the gain curve, what is an appropriate model to describe the population dynamics of the North Sea cod stock (without fishing)? Write the discrete form of this model.: The Logistic growth Model  Nt+1=Nt+RNt(1-Nt/K
What is N* (the equilibrium population) for this population (take a look at your class notes) if the carrying capacity of the cod stock is 400 and the discrete growth rate is 1.5?: RNt(1-Nt/K) = 0

1-Nt/K=0
-Nt/K=-1

-Nt=-K

Nt=K

Nt=400
First, create a graph showing Nt+1 vs. Nt for your equation in question 3. Draw the line on this graph where Nt+1 = Nt. At what population size do the two curves cross? What does this intersection represent?: They cross at (400, 400)

This represents N* or the equilibrium point in the model
In class, we talked about an experiment where we repeatedly drew numbered billiard balls from a hat, recorded the number on the drawn ball, then replaced them in the hat (a â€œwith replacementâ€ style of sampling). Explain in you own words why the â€œwi: If you donâ€™t add the billiard balls back into the drawing then the odds of drawing another ball increases because the pool which you are drawing from is constantly decreasing in size. The outcome of the first drawing provides information for the next drawing.
Consider a sampling scheme where a masterâ€™s student swims the length of a transect parallel to a beach and counts the number of tuna at 50 meter intervals. List three reasons why the fish counts recorded from this sampling might not be random.: a. The student may choose to sample at a certain angle as they are swimming because it is more convenient physically for them to do so.

b. The student probably wonâ€™t be able to sample at exactly every 50 meters because they will be focused on trying to actually swim. (It just isnâ€™t very precise)

c. Most likely the swimmer will be constrained in his/her attempt to swim by physical barriers etc. The swimmer will probably only be able to swim along a beach in areas which are suited for human swimming. Therefore this wouldnâ€™t be a random sampling of fish because the location of the observations is not random.
) What are three examples of demographic stochastic processes that affect baby birds in a nest?: a. Parental nurturing skills

b. Parentâ€™s luck at finding food

c. Parentâ€™s luck finding nest building materials
What are three examples of environmental stochastic processes that affect mosquitoes in a field?: a. Dry summer â€“ no water to breed in
b. Daily variation in wind/rain â€“ drowning/ unable to feed on prey
c. Lack of prey due to random poor growing conditions for preyâ€™s food
What are three examples of catastrophic stochastic processes that affect fish in a pond?: a. Severe Drought â€“ to the extent that the pond actually decreases in volume substantially
b. Hurricanes â€“could potentially clog up ponds with DWD (dead woody debris) or destroy them through flooding
c. Volcanic eruptions â€“ depositing ash and debris in to the pond
Think about the length of each fish counted along the transect from question 2. Would you represent the frequencies of these lengths as a discrete or continuous distribution? Why: It is a conintuous distribution because the values could be any value between some minimum and maximum set of values.
Think about the number of berries today on each bush on the south face of Cypress Mountain. Are the frequencies of these berry counts best represented by a discrete or continuous distribution? Why: They are best represented by discrete frequencies because the berries that are being counted in whole unit increments. If they were being monitored constantly by some automated machine and counted veeeerrryyy slowly as they were growing..that would be continuous.
What type (uniform or discrete) and class (uniform, normal, something else) of probability distribution would you expect if you were to record the sum of two six sided dice thrown 100 times? Sketch it out.: a. This is a discrete distribution because there are 6 distinct outcomes and no outcome can fall in between these predicted outcomes. (I am not sure what to do if the dice somehow lands on a vertex.)

b. This is definitely not a uniform or normal distribution. It is a something else which has discrete occurrences ranging from a few on the ends like 2 and 12 to a lot in the middle. The number six for example has multiple ways of adding up to equal it so it will have a higher frequency than 2 or 12.

It is a pyramidal distribution
What are the two potential risks to the persistence of the park rhinos?: a. Poaching
b. Reduced Carrying Capacity due to loss of habitat
The model presented in the Rhino paper described as a â€œstage classâ€ model (i.e., not an age class model). Why?: This is a stage class model because the classes being modeled are not based on age but based on the developmental stages (calves, subadults and adults) which are characteristic of the Indian rhino.

In addition, the oldest class could be any age over 7 and can die at any age interval
Explain the following difference equation in words: a. n1(t)=n3 (t-1)*S0*b3*f b.n2(t)=n2 (t-1)*S1 c.n3(t)=n2 (t-1)*S2+n3 (t-1)*S3: a) is the number of baby rhinos which survive to be age class 1 (rhinos only give birth at age 3 apparently)

b)The number of rhinos at age class 1 which survive to be age class 2

c) The number of rhinos at age class 2 which survive to be age class 3 and the number of age class 3 rhinos which survive in age class 3
Write this in matrix notation: a. n1(t)=n3 (t-1)*S3*b3*f b.n2(t)=n2 (t-1)*S1 c.n3(t)=n2 (t-1)*S2+n3 (t-1)*S3: n1(t) 0 0 b3*f*S3 n1(t-1)
n2(t) = S1 0 0 x n2(t-1)
n3(t) 0 S2 S3 n3(t-1)
How would you define carrying capacity in your own words?: Carrying capacity is the maximum number of organisms (at a given growth rate and survivorship) that the environment can support without incurring negative density dependent losses.

...and carying capacity is not always the equilibrium point

In addition, net growth is at zero
Why would you press the F9 key if you were using your model to make predictions?: The random numbers change and the trajectory fluctuates. You would press the F9 key in order to see how much the trajectory varied. It would be useful to identify the max and min points in the trajectories in order to develop estimates which encompassed all prediction possibilities of stochasticity effects.
Now one last change to simulate the effect of poaching... Have you added environmental or demographic stochasticity?: This is environmental stochasticity because fluctuations in loss due to poachers are experienced by all individuals as the pass into that stage class.
So what is the general effect of poaching predicted to be on the rhinos (according to study)?: Poaching appears to have the most detrimental effect on the rhino population compared with perhaps environmental stochasticity effecting carrying capacity.

It reduces the average equilibrium density which exaggerates the fluctations.
What is the point of REM 611?: To be able to help resource managers or become resource managers use models to:

Predict
Understand
Make Decisions
We do we use models for as Remmers?: Describe
Predict
Protect
Promote
Harvest
Understand
Communicate
What are some challenges?: Variation
Population
Communities
Ecological and Social Sciences are linked and can't be treated in isolation
What do Resource Managers Do:: a. Track populations
b. Interested in changes in size for the population
c. We want to know if pop numbers are going up or down
d. Tourism, pests, endangered species
Define a BIDE model: Birth and Immigration minus Death and Emigration determine population size
Geometric Growth Model Formula: Nt+1=Nt+RNt
What is R?: R is the growth rate or births minus deaths
Assumptions of the geometric growth model:: a. Discrete Growth
b. All individual are identical
c. They all contribute births equally
d. They have equal lambda
e. Lambda never changes
f. N can be non integer
g. No I or E h. closed population i. Geometric Growth: R= B-D
Analytical Solution for the geometric growth model:: Nt=λ^t*N0
Logistic Model Definition: Includes negative feedback from density dependence

Includes parameter for carrying capacity
Negative Feedback examples: a) Density Dependence - food loss

b) Interference - animal interactions, mates, feeding habits

c) Living Space

d)Aggression

e)Predators increase and concentrate efforts

f) Predatorslearn evasion techniques because there are more opportunities to learn

g) Immigration declines

h) Emigration increases
Logistic Model assumptions: Introduced density dependence

Assumes growth rate is negative linear function of Nt
What is the general shape of a logistic curve?: sigmoid
Coninuous Logistic Growth Model List the Analytical Solution and Graphical Solution: dN/dt

Eulers approximation :
Nt+1= Nt+(RN) h
What are 2 factors Malthus hypothesized influenced population size?: DI:

Random: fire, floods

Constant: hunting
Allee Effects: Problems faced by small populations such as finding mates, foraging efficiency, protection from predators and inbreeding problems
Stochastic: Introduction of random variables
Equilibrium: Population does not change over time

Continuous Model: dN/dt=0

Discrete Model: Nt+1=Nt, Nt+1-Nt=0
Stability of a population:: Tendency of system to return to its equilibrium state following a perturbation from that state
Global Equilibrium:: Returns to equilibrium point no matter how hard we push
Local Equilibrium:: Returns to equilibrium point, but this point is at a different position than the initial starting point
Resistance Equilibrium: resistance to leaving equilibrium
Resilience: speed with which system returns to former stat after a disturbance
What are 3 factors used in regulating a Fish Pond?: -increase food

-decrease death rate
(use less effective tackle)

-increase size of pond
Describe a fishing pond with regulation.: Loss
Growth
EQ

Limitation: (DD)
-Higher death rate will have a steeper line
-Density DD growth and DD loss

-But there is a point where growth balances loss

-Higher r makes the curve pointer
Describe fish pond with low death What happens if add food? How about changing the size of pond?: changed tackles

if d is low alreadyâ€¦managing for d doesnâ€™t do much

population is insensitive to d

try to add foodâ€¦nothing happens

Change size of pond! Very Responsive
Describe a fish pond where there is initial high death What happens when death decreases? Changing pond size? Changing food?: Decreasing death has a huge impact

Changing pond size had negative effect

Increasing food didnâ€™t help (K)
Limitation Formulaic Representation: Nt+1 â€“ Nt= 0

and

dN/dt=0
Describe Discrete Testing for Equilibrium (Geometric): Nt+1=Nt+RNt where R is discrete per capita growth

If Nt+1-Nt=0 then

Solve for Nt ⬦Nt+1-Nt=RNt

0=RNt

Nt=0
Numerical Testing for Equilibrium (Geometric): find N*

Set up model on computer

perturb system (move little up or down⬦local stability)

A lot up or a lot down is for global stability
Rickor Diagram â€“ Graphical Approach (Geometric) Describe derivation -: Nt+1=Nt+RNt

Nt+1=(1+R)Nt +0

This is a line Y= mX+B

R=birth-deaths

If R(t) then the slope is >1
Describe the Discrete Logistic Equilibrium Approach (Analyitical): Nt+1=Nt+RNt(1-Nt/K)

Nt+1-Nt=0 Equilibrium

Nt=1-Nt=RNt(1-Nt/K)=0 Now solve for Nt

Nt=K But we knew that already
Describe the Rickor Diagram Logistic: Nt+1=Nt+RNt(1-Nt/K) Solve for line

Nt+RNt-R(Nt^2)/K

This is the equation for a parabola
Is a two point stability cycle stable?: It is not globally stable
Is there a graphical way of determining eq for a Continuous Models ?: No, only simulation or anylitical
Neighborhood Stability Analysis (Only local stability test for) Formula:: dN/dt=F(N)
Perturbation over time for a continuous model Formula:: So Perturbation over time= N(t)=N*+x(t)
So Perturbation over time= N(t)=N*+x(t) (in a continuous model) Calculus formula:: How does x change over time? dx(t)/dt?
For a continuous model: a=âˆ‚F(N*)/âˆ‚N* in the formula -> dx(t)/dt=ax(t) Describe the outcome of a:: a<0, then locally stable and perturbation is lessening over time

a>0, than perturbation is increasing and is not stable

a=0, then never changes over time and is neutrally stable
Paradox of Enrichment Describe:: lets assume this can be modeled with logistic discrete model

Outcome is that population becomes less stable when increasing population

(Comes to a two point stable limit cycle)

When r gets higher and higher we get more and more limit cycles until eventually there will be chaos
regulation: forces that bring population back to their equilibrium

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