Done so far Flashcards
possible ways to add more complexity or reality to exponential or growth models
- different forms of density dependence (allee effects)
- time lags
- incorporate species interactions (eg effects of competitors, predators, mutualists)
per capita growth rate is fastest when… what is an exception to this?
population is near zero; sometimes more density may be beneficial
what are Allee effects?
negative effects of low density, arising from social benefits such as mate finding, group living, group defence
meerkats
cooperate to avoid predators and rear young, so their populations require a minimum population density to grow
when Allee effects are in force
- populations may fluctuate between carrying capacity, K, and another, lower limit
- dropping below the lower limit goes to extinction
- very important in conservation
age-structured populations
- exponential and logistic models of population growth treat all individuals in a population the same
- but in real populations, not all individuals have the same probability of giving birth or dying
- fecundity and survivorship depend on age
- how these depend on age varies among species; species have different life history strategies
key components of a life history strategy include
lifespan, the timing of reproduction, number of offspring, and parental investment in offspring
typical life history for many plant and animals
- start life at small size
- grow for a period without reproducing (for resource accumulation)
- when have enough resources, become mature, start spending resources on reproduction
- organisms show various lifestyles after sexual maturity
- some expend all resources at once, see spread them out
- need to consider age structure of population to better predict population trajectories
elephants
low fecundity
long lifespan
late 1st reproduction
big investment in each individual offspring
pika
high fecundity
medium lifespan
fast first reproduction (within 1st year of life)
1-13 babies per reproduction cycle
salmon
very high fecundity
medium lifespan
late first reproduction
return to natal rivers at end of lives to have offspring then die right after
female can lay 1000s of eggs when she spawns
variation in fecundity and survivorship with age is summarised by
life tables of age-specific rates
life tables have important implications for
- evolution of life histories
- conservation of populations
- understanding the changing structure of human populations (human demography)
age-sex pyramid
males left, females right, height of bar Indicates how many individuals there are of that population
demographic transition undergone in Canada
pyramidal shape -> stable age structure with similar number at each age class
which sex is usually only used in age structures?
females - these are assumed to invest the most time and energy into rearing offspring, and so limit the amount of children
age-class intervals
- arbitrary units of time chosen to give a reasonable number of age classes for the organism in question
- for microbes, minutes to hours
- most insects, weeks
- most mammals and birds, years
- humans, typically 5 year intervals
life tables
- data that summarise the life events that are statistically expected for the average individual of a specified age in a population
- age of death
- age and timing of reproduction
- for modelling, these are treated as constants
- usually consider females only
age classes denoted by
subscript x
lx =
probability of being alive at age x
l0 =
1.0 by definition
survivorship curve
graph of lx vs x
lx necessarily declines with
x
shape of lx curve
characteristic of species
draw and describe types of survivorship curves
type II - if mortality is constant with age, you will get an exponential decline; usually graph lx curves as log plots, where type II is a straight line
type I - individuals survive really well until middle age
type III - really high mortality early in life, but if you make it into adulthood you have a high chance of surviving
real shapes of survivorship curves + example diagram
more complex
real human data example from Statistics Canada
fecundity schedules
- mx (or bx) = number of daughters born to a female of age x during the interval x to x+1
- shape of mx curve is characteristic of species
- reproductive period usually preceded by resource-accumulation phase
- fecundity-survivorship trade-off = cost of reproduction
net reproductive rate
- average (expected) number of daughters a female has in her lifetime = R0
- R0 = Σlxmx
why does net reproductive at work?
Σmx would be the total number of daughters produced by a mother who doesn’t die earl; multiplying by lx discounts expected production by the probability that some mothers do die early
R0 is like
λ, but in time units of one generation rather than one time interval
in epidemiology, R0 is
the average number of secondary infections that a single infection gives rise to
generation time, T
average age at which a female gives birth
T = Σxlxmx/R0
- this is a formula for weighted average. x is a female’s age; multiplying x by lxmx weights x by how many offspring are produced at that age; dividing the sum of the weighted x’s by the total lifetime production of daughters (R0) gives a weighted average that specified when a female gives birth, on average
relationships among R0, λ, r
- these parameters indicate the factor by which a population changes during a discrete interval of time, but those intervals are different
r = ln(R0)/T = ln(λ)
how are growth rate and fitness related?
generally, organisms with higher growth rates have higher fitness
why aren’t all plants annuals? why aren’t all mammals mice? why aren’t all lives short and fast?
constraints and trade-offs: reproduction is costly. longer pre-reproductive periods allow time to accumulate more resources
plant life history category table
example of obligate semelparity
when does natural selection favour semelparity?
when reproductive output is increased by accumulating resources for longer, for example if:
- reproductive output depends strongly on size
- in plants, massive flower/fruit displays attract more beneficial animals (pollinators or seed dispersers)
- or massive seed crops satiate seed predator populations, allowing more seeds to go uneaten
semelparous fish lay larger eggs, but only if
they grow large enough
example of extremely synchronised semelparity
- bamboo
- long lived
- long reproductive period when growth of offspring is highly synchronised
- thought to satiate the bamboo seed predators so the predators can’t eat all of the offspring
advantage of synchrony
infrequent pulses of reproduction = predator satiation tactic
iteroparity plus local synchrony
masting
eg quercus douglasii
K strategy vs r strategy
Outcome of competition
hurts both species
Outcome of predation
benefits predators, but hurts prey
Outcome of host-parasite and plant herbivore interactions
same as predation - positive and negative
Outcome of mutualism
helps both species
Interactions between species are often classified by
their outcome (+ or -)
Two main foci of study in ecology and evolution of species interactions
- population dynamics and effects on community structure (how species interactions affect these two things)
- evolutionary dynamics (adaptation and co-evolution)
Intra-specific competition
competition among the members of the same species (i.e. among conspecifics) for resources
Inter-specific competition
competition among members of different species (ie among heterospecifics) for resources
Scramble/exploitative competition
depletion of a shared resource
Contest/interference competition
direct interactions, such as battles over territory
Give an example of interference competition
- Invasive Argentine ants fight a harvester ant in California
- Invasive ants (superior competitors) often drive down populations of native ants
Exploitative competition
- two species do not need to directly interact or even to be active at the same time to compete
- if one consumes a resource, leaving less resource for the other, then they compete
Example of exploitative competition
squirrels and birds, and bird feeders
- squirrel eats food from the feeder and leave no seeds left from the birds
- squirrel is competing successfully with birds by consuming a lot of shared resources
model for inter-specific competition for resources
Lotka-Volterra equations for two species competing for resources
- is a simple outgrowth of logistic equation
- logistic already has a breaking term for intra-specific competition
- Just add a second braking term for
inter-specific competition
give the 4 steps for arriving at the Lotka-Volterra model from a logistic model
- Start with the logistic model for population growth
- Rewrite the logistic model with subscripts to indicate species 1
- Add a term to show effect of species 2 on species 1
- Write matching equation for species 2
give the equation for Lotka-Volterra model
α(ij) =
α(ji)
per-capita effect on i by j
per capita effect on j by i
=> competition coefficient
describe the competition coefficients (α’s)
- fixed for a particular pair of species
- α(12)N(2) converts individuals of species 2 into an equivalent number of individuals of species 1
- eg a squirrel can eat a lot more seeds than a sparrow; a measures how many sparrows-worth of seeds a single squirrel eats
four possible equilibria outcomes of Lotka-Volterra competition
- the two species may stably coexist
- species 1 may always win (N1 = K1, N2 = 0)
- species 2 may always win (N2 = K2, N1 = 0)
- identity of winner may depend on starting N’s
meaning of Equilibrium for Lotka-Volterra competition
N’s are no longer changing
what do the outcomes of the Lotka-Volterra competition depend on?
values of K’s and α’s
coexistence requires
both species to inhibit their own growth more than they inhibit each other’s
define equilibrium
- for a population: size not changing over time (dN/dt = 0)
- for a community: a community not changing over time (in a strict sense: all populations in a community at equilibrium. more generally: constant species composition)
define stability
the ability of a system to return to equilibrium following a perturbation or disturbance
define coexistence
occurs when two or more species have non-zero population sizes at equilibrium
Principle of competitive exclusion
- Lotka-Volterra predicts that for two species to coexist, competition between species must be weaker than competition within a species
- in other words, two species can’t compete too intensely (i.e. overlap too much in resource use/niche space), or one will outcompete the other
- This idea is very old: “As a result of
competition two similar species scarcely ever
occupy similar niches” (Gause 1934) - Or: “Complete competitors cannot coexist”
(Hardin 1960)
Character displacement
coexisting similar species evolve differences to minimise effects of competition on their fitness
- eg Darwin’s finches and beak size
- when finches live on same island, beak size becomes different so that they can eat different sized seeds
Paradox of the plankton
In aquatic ecosystems like lakes and oceans, phytoplankton species coexist in high diversity, even though they:
- Compete for the same basic resources (e.g., light, nutrients like nitrogen, phosphorus, etc.).
- Share similar ecological niches.
This coexistence defies the competitive exclusion principle: How can so many plankton species coexist in an environment with limited resources?
Paradox of the tropical forest
- hundreds of species of trees living in very small areas, despite having the same niche
- how is this possible?
- either every species has a distinct niche or something prevents competitive
exclusion from driving species extinct - this is subject of intense study/debate
how do Lotka-Volterra models relate to the real world?
Experiments by Gause (1930’s)
studied competition among protozoa
in artificial culture vessels; saw both stable
coexistence and competitive exclusion
Gause’s famous competition experiments with Paramecium species in lab culture - draw graphs
How are competitive effects manifested in nature compared to the lab?
- in nature, competitive exclusion is less likely to go to completion
- nonetheless, competition can drastically affect abundances and alter distributions in space
- Biological effects interact with physical
effects: different outcomes in different
environments
Connell, 1961: Field experiments with two barnacle species in the marine intertidal zone
- Zone upper limits set by desiccation
- Lower limits set by competition for space on the rock
- Competition is asymmetrical
- Remove Balanus, Chthamalus extends its distribution down (distribution limited by competition)
- Remove Chthamalus, Balanus does not extend upwards; not competition, but simply can’t tolerate the conditions at the top of the rocks
Resolving the paradox of the
plankton
Lotka-Volterra models too simple, ignore too
much reality, including:
- Most real communities are not at a
competitive equilibrium
- Real populations are kept below carrying
capacity by weather, disease, predators
- Real conditions fluctuate, favouring different species at different times (or in different places)
Scaling up from two populations
to ecological communities
- Competition can affect which and how many
species occur in an ecological community,
which ecologists call community composition
and species richness, respectively - Competition is generally expected to
decrease species diversity (e.g., if a superior
competitor excludes other species) - It is a real challenge to scale up from simple,
species-poor systems (e.g., two Paramecium
in lab cultures) to complex, species-rich
systems (e.g., a whole tropical rainforests)
population size symbol
N
population density symbol
N/area
why do we care about understanding population size, N?
- natural resources management (eg size of fish stocks in the ocean, abundance of outbreaking insect pests in forests)
- conservation: population decline of a species
- health: monitoring populations of viruses or bacteria in humans
- understanding and predicting human population growth
- basic science question of what limits population growth
Population declines in Myotis lucifugous bats due to white nose syndrome (WNS)
- novel pathogen (fungus) emerged in bat population, causing a disease caused white nosed syndrome and really high mortality.
- steep decline in no of over-wintering bats
HIV population dynamics in humans - draw graph of CD4 cells over time
Malthus’ essay on population growth
in 1798, Malthus published an essay on the principle of population, arguing that the human population cannot grow faster than food production
Paul Ehlrich
published The Population Bomb, arguing that explosive growth in the human population would have catastrophic social and environmental consequences
how is the human population expected to change in the future?
demographers project that human population is soon going to peak, then fall dramatically (depopulation)
goals of most population models
- predict the trajectory of population growth through time, i.e. N as a function of t
- how many individuals are in the population now? Nt
- how many individuals are in the population one step later? N(t+1)
- so the general model is N(t+1) = fN(f)
- challenge: choosing simple but realistic parameters for f
when using differential equations, time steps are
infinitesimally small: use concept of limits and calculus; growth is smooth; best suited for species with continuous reproduction
when using difference equations, time steps are
discrete units (days, years, etc); use iterated recursion equations; growth is stepwise and bumpy; best suited for episodic reproduction
two types of time step approaches
continuous-time and discrete-time
how do we pick between the two time-step approaches?
different organisms might be better fit by one or the other
simple bookkeeping model: how can N change from Nt to Nt+1
D = number who die during one time step
B = number born during one time step
E - number who emigrate during one time step
I = number who immigrate during one time step
Nt+1 = Nt - D + B - E + 1
what variables can we consider to be equivalent?
- birth and immigration (ie individuals added to the population)
- death and emigration (ie individuals that disappear from the population)
geometric growth model
- assume no immigration or emigration
- treat birth and death during one time step as per capita rates that are fixed constants
- then, population changes by a constant factor each time step: N(t+1) = λNt
- λ is a multiplicative factor by which population changes over one time unit = ‘finite rate of increase’
- λ = Nt+1/Nt
if λ>1,
birth exceed deaths and population grows
if λ<1
deaths exceed births and population shrinks
N1 =
λN0
N2 =
λN1 = λλN0
N3 =
λN2 = λλN1 = λλλN0
so how can geometric growth be generalised
Nt = N0λ^t
exponential growth
- instantaneous, fixed per-capita rates of birth and death (b and d)
- instantaneous, per-capita rate of population change = b-d=r (a constant)
- r = intrinsic rate of increase
- differential equation is dN/dt = rN
- this model is exponential growth
draw a table comparing discrete-time and continuous time growth models
find the relationship between r and λ
lnλ=r
regardless of which model is adopted, the important consequence is the same
- in both models, the growth rate (λ or r) is a constant that simply reflects biology
- but a constant positive growth rate produces a population growth size that is not constant, but rather exploding in an exponential way
all species…
- have the potential for positive population growth under good conditions (λ>1.0, births exceed deaths)
- have the potential for negative population growth under bad conditions (λ<1.0, deaths exceed births)
- but no species has ever sustained λ>1 or λ<1 for a long period
why is exponential growth a bad model of reality over the long term?
- some factors use tend to keep populations from exploding or going extinct
- two kinds of factors may be acting: density dependent regulation (growth depends on N) or density-independent reduction
how can we model the classically, density-dependent growth?
the logistic equation; an exponential growth with a new term added for brakes
dN/dt = rN(1-N/k)
use bacteria as an example of two types of growth
The logistic braking term models… (draw graph)
the simplest form of density dependence
K
carrying capacity of the environment
logistic trajectories are truly
S-shaped only when
starting from low numbers
label an N vs t graph
logistic model pros
- Mathematically tractable model of intraspecific competition for resources
- Simple (only one extra parameter, K, beyond exponential)
- Can be expanded to consider multispecies competition
logistic podel cons
- Too simple: specifies one particular kind of
density dependence - Always a gradual approach to carrying
capacity - In reality, density-dependence is likely
to be non-linear, may see overshoots of K
interactions in which one organism consumes all or part of another
- predation/carnivory:
- prey is killed
- predator generally larger than prey
- multiple prey individuals per predator - grazing/herbivory:
- plant survives, usually - parasitism/disease:
- host may or may not survive
- host generally larger than parasite
- multiple parasites per host
Brood parasites
- some birds lay eggs in the nests of other bird species, avoiding the costs of parental care
- often involves brood mimicry, in which parasite eggs evolve to resemble host eggs
Lotka-Volterra models for predator-prey interaction tend to..
cycle
- similar to competition models: two differential equations
- predict couples, lagged population cycles
draw diagram
most common lab result for cycles in predator-prey interactions
predator and prey do not coexist, interaction is unstable.
For Huffaker to achieve 3 cycles was a triumph of persistence
most famous predator-prey cycles outside the lab
Lynx and Hare
why are Lynx-hare cycles not simple Lotka-Volterra predator-prey cycles?
additional factors probably include:
- heavy browsing degrades quality of plant food available to hares - hares may also be cycling with food plants
- social stresses in overcrowded hare populations
give an example of a disease cycle
measles before vaccination
- number of measles cases between 1944 and 1966 cycled
- in outbreak years, where there are many infections, most people would recover from the infection and become immune
- after an outbreak year, the measles couldn’t infect many new hosts
- once enough babies were born, the measles would spread again
- cycles driven by no of susceptible and immune humans in the population
describe how COVID cycles in humans
- waves of COVID cases were thought to be caused by human behaviour changes
antagonistic co-evolution
- coevolution = reciprocal adaptation
- prey evolve defences; predators evolve counter-adaptations to overcome defences
- frequently described as an ‘arms race’
- key to the ‘red queen hypothesis’
red queen hypothesis
species must constantly adapt, evolve, and proliferate in order to survive while pitted against ever-evolving opposing species
Garter snake and rough-skinned newt
- Pacific newts (Taricha) make a poison called tetrodotoxin (TTX) that is extremely toxic to many predators
- garter snakes (Thamnophis) have evolved resistance to TTX
in populations where newts are more toxic, snakes are more resistant to TTX
Life-dinner principle
- predator–prey interactions are characterised by unequal selection pressures operating on the participants.
- one party is ‘running for their life’ and the other merely for their dinner
give examples of victim defences
- prey morphology, chemistry, behaviour
- plant secondary chemicals
- human immune system
- Daphnia ‘helmets’
inducible defenses
defences turned on in response to threats or attacks
Impact of competition on biodiversity
Competition tends to decrease biodiversity;
superior competitors exclude inferior
competitors
How does predation affect
species diversity in a
community?
Classic example is Paine’s
Pisaster (sea star) experiment in the rocky
intertidal
- Pisaster predation prevents mussels from competitively excluding other species in rocky intertidal communities, maintaining biodiversity
How do predators and parasites influence biological invasions?
- invasive species achieve high population sizes and have negative effects on the communities they invade
- enemy release hypothesis: invaders’ impacts result from having fewer natural enemies (predators, parasites, or pathogens) in their new range, compared to their native range
example of how predators and parasites influence biological invasions
fungal and viral pathogens of 473 plant species that have been introduced from Europe to North America is much lower in naturalised environment than native environment
describe the life cycles of parasites
- some parasites have a single host species - direct life cycle
- many parasites require two or more host species to complete their life cycle = complex life cycle
- the parasite that causes malaria passes through two hosts, a mosquito and a human
vectors
hosts that transport parasites to their next shot
zoonotic diseases
diseases transferred between animals and humans
host species of zoonotic diseases
reservoirs
what affects parasite abundance and transmission?
distribution, life history traits, and behaviours of hosts
How is disease risk to humans, livestock, or wildlife affected by the broader ecological community?
Competing ideas:
- Dilution effect: for diseases that infect many hosts, host diversity can “dilute” disease risk to humans or animals
- Amplification effect: more host or vector species can support larger populations of disease-causing organisms, increasing risk to humans or animals§
Amplification effect: malaria
- The malaria-causing parasite Plasmodium falciparum is vectored by many species of Anopheles mosquitoes
- study region in Kenya examined four mosquito species: A. arabiensis, A. funestus, A. gambiae, and A. merus
- positive relationship between mosquito species richness and prevalence of malaria in Kenya schoolchildren
Latitudinal gradient in species richness
Species richness, or biodiversity, increases from the poles to the tropics for a wide variety of terrestrial and marine organisms
Latitudinal gradient in human pathogen species richness
On average, tropical areas harbor higher pathogen species diversities compared to more temperate areas.
