Ecology Notes PDF

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Population Growth 11.07.18  ***Populations grow by multiplication***  Geometric Growth Equation: Nt+1 = 2 x N t. Geometric growth occurs when reproduction happens at discrete time intervals (ex. Annual plants, cicadas, etc.).  Alternate Equation for Geometric Growth: Nt = t N 0 Population at time t = finite rate of increase (growth rate) x initial population size. This equation creates a J-shaped curve.  If is 1, population is stable. If > 1, population is growing. If 0 < < 1, population is shrinking. Geometric growth assumes individuals reproduce in synchrony (generations do not overlap).  Exponential growth occurs in populations with continuous reproduction and overlapping generations (ex. humans). Equation: Nt = ert x N0 where e is a constant raised to the finite rate of increase, r. Exponential growth creates a J-shaped curve.  If r = 0, lambda is 1 and population is stable. If r > 0, lambda > 1 and population is growing. If r < 0, 0 < lambda < 1 and populations is shrinking.  Geometric and exponential equations describe the same growth. Exponential shows a more cohesive J-shaped curve while geometric shows a set of points in a J-shaped curve.  Short Answer Problem: - If population was 185 in one year and then 198 the following cycle, what is the geometric rate of increase? (Nt = t N 0) 1.07  - Assuming growth rate does not change, what will the population size be two cycles later? 227  - What is the net reproductive rate for this population of cicadas? 1.07 because with a geometric population, lambda is equal to the net reproductive rate   There is a finite amount of resources in the universe, therefore there is only a certain number of individuals our planet can support.  ● Ex. You are growing a bacteria in a closed jar. What will happen to the population?

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Population will start to grow. Resources and space will be depleted, aggression/competition will increase. Carrying capacity will be reached, population will start to level o and will continue at that same size.

 Logistic Growth v. Exponential Growth: - [dN/dt = rN] is the exponential growth equation. [dN/dt = rN (1-N/K)] adds a discounting term to account for density. This discounting term explains the bend in the exponential curve as carrying capacity is approached (exponential growth is density-dependent). - N is current population size, r is finite rate of increase under ideal conditions, K is carrying capacity. - In exponential growth, r is constant. Logistic growth assumes K is constant.  Population Regulation: - Leveling o suggests that extrinsic forces limit growth as population size (and density) increases. - Births, deaths, and dispersal (emigration and immigration) all contribute to change in population size.  Density-Dependent Population Regulation: factors that might decrease births, increase deaths, or increase dispersal out of an area when population density increases. - Ex. availability of food, predation, disease and migration, parasites. Because density-dependent factors respond to population density, they can regulate populations.  Density-Independent Factors do NOT regulate population size, but they do aect the population.

SimUText Assignment 18: Competition (Sec 1: 1-5, 11-12; Sec 3: 1-26)  If resources are unlimited, populations grow exponentially and competition does not occur. When resources are limited, similar organisms compete for those resources. This impacts both individual organisms and whole populations. Resources are always limited over the long term so living organisms are almost always competing for something.  All living organisms need certain resources to live and reproduce. Each organism needs energy (photosynthesis, consumption of other organisms, consumption of detritus). Each organism also needs nutrients like nitrogen/phosphorous and some require complex molecules like amino acids.  Populations will grow rapidly if they have unlimited resources. Even if resources are temporarily abundant, the resulting population growth will eventually lead to resource depletion. When their life-sustaining resources are limited, organisms are forced to compete with each other. Sometimes those competitive interactions will be direct while other times they will be subtle.  The competitive exclusion principle states that no two species can coexist in exactly the same ecological niche. As a result, if two species are using exactly the same resources, one of them is expected to win the competition and to drive the other species to extinction in that location.  A species' fundamental niche is the complete set of environmental conditions in which the species can potentially survive and reproduce. This includes both abiotic factors like temperature and precipitation as well as biotic factors like food availability. However, unlike the realized niche, the fundamental niche is not restricted by interactions with other species.  A species' realized niche is the set of environmental conditions in which the species is actually found in nature. It is a subset of the species' fundamental niche; the realized niche is further restricted by biotic interactions like competition, predation, and parasitism.  Competition is the interaction between two organisms that use the same limited resource. Both organisms tend to be negatively aected by this interaction and can suer increased mortality and reduced reproductive capacity because of it.

Competition can also be an important driving force of adaptation, however, and may ultimately influence how species evolve.  Competition can occur between members of the same species (intraspecific competition) or between members of dierent species (interspecific competition). Plants usually compete for physical space, water, light, and soil or water nutrients; animals often compete for food, territory, and mating partners.  Parasitism is a form of symbiosis where one organism lives in close association with another and gains advantage from doing so. The parasite gains from this relationship whereas the host is harmed in some way.  It is easy to find situation where competitive exclusion does not operate; there are also some theoretical problems with the definition of a niche but both concepts are very helpful in ecological research.  Ecologists quantify the relative competitive strengths of dierent versus same species competitors with competition coecients.  Competition coecients represent the per capita eect on the growth of one species (species 1) of another species (species 2) and are symbolized by the Greek letter alpha, α, and two subscripts. Thus, the competition coecient describing the eect on species 1 of species 2 would is written as α12 while the competition coecient describing the eect on species 2 of species 1 would be written as α 21 . By convention, the first subscript indicates which species' growth is being reduced while the second subscript represents its competitor.  The logistic growth equation with N1  K1 ) 

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NO COMPETITION: dN 1 /dt = rN 1 (1

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EQUIVALENT COMPETITOR: dN 1 /dt = rN 1 (1

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ANY COMPETITOR: dN 1 /dt = rN 1 (1

N 1+N 2  K1 ) 

N 1+α 12N 2 )  K1

 The equations for when any competitor is present is called the Lotka-Volterra competition equation. It predicts how the population size of two competing species will change over time (you can use K2 and N2 in the equation for the second population). 

Ecologists use a phase plane graph to plot the population sizes of both species against each other. A single species has reached its equilibrium population size when its population is no longer growing or contracting. Mathematically, the equilibrium population size is reached when dN/dt = 0. For a species without competitors or predators, the equilibrium population is equal to the species' carrying capacity. In other words, for this single species, dN/dt = 0 when N = K. Technically, it is also true that dN/dt = 0 when the population has gone extinct, i.e., when N = 0, but this is usually a less interesting equilibrium.  When two populations have reached their equilibrium size, they stop growing and all future points are plotted on top of each other. When no changes occur in population sizes over time, a steady state point has been reached.  Sometimes a steady state is globally stable (there is one state the system will reach no matter where it starts). A phase portrait, plotted on a phase plane, is a type of two-dimensional graph that describes how a dynamic system will change over a short period of time, given dierent initial values for the graphed variables. It depicts (usually using arrows) trajectories of change for each combination of the two variables.  The isocline of zero population growth line for a pathogen indicates the condition under which the pathogen population (N1) will not grow or shrink (N2 for N1 and K2 for K1 can be substituted to measure the zero population growth of the probiotic). N 1 =  α12N 2 + K 1   When two isoclines intersect and all phase portrait arrows point toward the intersection, this indicates that the system has a stable equilibrium: no matter what the initial population sizes are, the two bacteria strains will ultimately coexist. Population sizes of both species will move toward the intersection of the isoclines and then remain at those sizes indefinitely.  Increasing the carrying capacity (K2) of the probiotic shifts the location of the probiotic’s isocline farther and farther from the origin. When K2 is very large, the probiotic isocline lies entirely outside the pathogen’s isocline. All of the arrows of the phase portrait point to a steady state on the y-axis where the pathogen population size is zero.  If the two isoclines don’t intersect coexistence is not possible and the species who isocline lies furthest from the origin will eliminate the other from the system.

Changing α12 alters the location of the pathogen’s isocline. As α12 is increased the y-intercept shifts further toward the origin. When α12 is large enough the pathogen’s isocline is so close to the origin that it no longer intersects the probiotic’s isocline. This indicates that the probiotic will outcompete the pathogen and eliminate it from the system.  If a process or system is at an equilibrium, its overall behavior is not changing over time. If the system is at an unstable equilibrium, small perturbations (changes to the system) will upset the balance and the system will not likely return to the same state. A special case of equivalent competitors exists when α12 = α21 and K1 = K2.  Stable coexistence results when intraspecific competition is stronger than interspecific competition (when K2 < K1/α12 and K1 < K2/α21). Smaller intraspecific competition leads to a smaller carrying capacity because individuals of the same species compete for the same resources.  The strength of competition between species often reflects how similarly they use resources; species are less likely to coexist when their resources are more similar. When two species have strong competitive eects on each other, one of them will likely be driven to extinction. *Interspecific competition is stronger than intraspecific competition*                     

SimUText Assignment 19: Predation (Sec 1: 1-2; Sec 2: 1-14; Sec 3: 1-23) The organism that a pathogen or parasite lives in or feeds on is its host. A parasite may require several hosts of dierent species to complete its entire life cycle. The definitive host is the host in which the parasite reaches maturity and undergoes sexual reproduction. The other hosts are intermediate hosts, and often play an important role in dispersing the parasite to places where it can reinfect new definitive hosts.  Predation, parasitism, and herbivory are forms of exploitation and can be represented as (+/-) interactions. One species gains (+) and the other loses (-). A species can also be neutral (0) in an interaction with another species.  Types of Species Interactions - Competition (-/-) : both species do worse in the other’s presence than they do if the other species is absent. - Amensalism (0/-) : one species is unaected by the interaction while the other is harmed. - Exploitation (-/+) : one species benefits and one is harmed. - Neutral (0/0) : two species inhabit the same area but don’t have a positive or negative aect on one another. - Commensalism (0/+) : one species benefits from the interaction while the other is unaected. - Mutualism (+/+) : both species benefit from the interaction.  Exponential Growth: N (t + 1) = λN(t) where N(t) is the size of the population at time t, λ is the number of ospring produced per individual each unit of time, and N(t + 1) is the size of the population one unit of time after the present time.   If the unit of time is very small, the equation can be rewritten as dN/dt = rN (EXPONENTIAL GROWTH EQUATION). dN/dt is how fast the population size is changing and r represents the population’s intrinsic growth rate (growth rate per individual when resources are unlimited).  The per capita growth rate is the rate at which additional individuals are being added to the population, per individual already in the population.

dN prey

Prey growth with predation: dt  = r prey x N prey - aNpreyN predator  Encounters with predators should cause the prey population to grow less rapidly. The more predators there are, the more encounters there will be and the slower the prey population should grow. The rate at which predators encounter prey also depends on how many prey are available. More prey available means more prey consumed.  The encounter term (N preyNpredator) increases with more predators or more prey. Because this term is negative in the equation, as the number of predators/prey increases, prey growth will decrease and vice versa if the predators/prey decrease.  The searching eciency parameter (a) represents the variation in how fast predators can find/attack/kill prey. Larger values of a reflect more ecient predators. aNpreyNpredator is the consumption rate.  Predator growth (partial conversion) with death:

dN predator = dt 

abN preyNpredator - mN predator

 The prey consumption rate (aN preyNpredator) defines how quickly predators reduced prey. This term represents how much the predators are eating so it also determines how quickly the predator population will grow. The predator population should grow in proportion to how much they eat but it implies that consumption of one prey individual produces one predator individual which isn’t necessarily the case.  A predator has to eat many prey before it accumulates enough resources to have ospring. A conversion factor (b) is used to convert the encounter rate into the number of new predators produced. Predators die and random death is modeled as a proportion of the predator population dying in each unit of time (negative m for mortality).  Adjusting predation and prey growth rates cannot eliminate cycling and cannot cause prey to go extinct. If the prey population is large and the predator population is small, the predator population will increase. If the predator population is large, the prey population will decrease. If the prey population is small and the predator population is large, the predator population will decrease. If the predator population is small, the prey population will increase.  Growth of the predator population lags behind growth of the prey population because the abundance of prey stimulates predator population growth. The relationships

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between these growth are presented as continuous up and down cycles. These cycles are regular and even with predator cycles following prey cycles.  The Lotka-Volterra equations all listed above are deterministic: the values of the parameters and the initial population sizes determine exactly what will happen. A stochastic model includes randomness and variation.  Predators can reduce the number of prey directly by killing individuals. But predators can also aect prey indirectly, when prey respond to the risk of predation by altering their behavior. The concept of the ecology of fear integrates the behavioral changes that can result from predator-prey interactions with the numerical changes. An example of the ecology of fear is illustrated by behavioral cascades.  Behavior cascades are a specific type of top-down eect in which the impacts of adding or removing a predator aects primary producers. They dier, however, in that the population size and/or biomass of the herbivore is not necessarily aected by the predator. Instead, when the predator is present, it alters the behavior of the herbivore to such a degree that the herbivore can no longer exert enough grazing pressure to limit plant growth.  A two-dimensional space showing one state variable plotted against another state variable is called a Phase Plane. Each timestep, a point in the Phase Plane is plotted at the intersection of prey and predator population sizes (Nprey, Npredator). If the prey and predator populations are both small, a point will appear in the lower left part of the phase plane. If they are both large, a point will appear in the upper right, and so on. The trajectory of this graph shows how the two populations change relative to each other.  Each cycle of prey v. predator can be characterized by two numbers. The amplitude of the cycle is the height dierence between the top and the bottom of each cycle. The period is the amount of time from one peak to the next in the cycle.  If there were phase plane plots for 2 dierent predator-prey systems, you would be able to see the dierence if they had cycles with the same period (time required to experience one peak and one trough) but dierent amplitudes (population size dierence between peak and trough). 

You would not be able to see the dierence between cycles of the same amplitude but with dierent periods. The location and the width/height of the phase plot circle corresponds to the amplitudes of the cycles on the time plot.  When predator and prey populations are modeled using the Lotka-Volterra equations, the cycle through the same peaks and valleys in size, time after time. Because there is no dierence in the amplitude of the cycles, each cycle lies right on top of the previous one in the phase plane.  The Lotka-Volterra predator-prey equations are neutrally stable. This means that changing the initial population sizes simply causes the population cycles to proportionally adjust, returning to their new sizes again and again.  The arrows in the phase portrait indicate whether the predator and prey populations will increase or decrease in the next timestep given dierent starting positions in phase space. If the initial combination of predator and prey population sizes is at the arrow’s origin, the prey population will decrease and the predator population will stay exactly the same size in the next timestep.   

 There is one particular combination of initial predator and prey population sizes that generates such a small, tight trajectory that it plots as a single point. When the predator and prey populations start out at this combination of sizes, there will be no cycling because neither population will change in size over time.  Assumptions of the Lotka-Volterra predator-prey model: - There are no eects of crowding for either the prey or predator (there is no density dependence in growth rate). - All predator and prey individuals are equally likely to meet any of the others (populations are evenly spread out and well mixed). - The prey species is the only food source for the predator. - The predator is the only significant cause of death for the prey. - Each predator individual can catch and eat prey individuals instantly (there is no handling time). - There is no immig...