Temperature-dependent population models of aquatic plants Flashcards
demographic stochasticity
how populations randomly change with time (e.g, births, deaths)
deterministic models in terms of vital rates
vital rates such as birth, death, migration are constant and unchanging over time
why is it important to account for demographic stochasticity on models?
- it alters coexistence patterns
- disease duration time is increased
- metacommunity spatial synchrony is reduced (metacommunity = communities connected by dispersion of interacting species)
- increased extinction risk
How can we calculate the effects of demographic stochasticity?
- add white noise to differential equations (from deterministic to stochastic differential equations = SDEs)
- embedding SDEs into probability distributions
- stochastic simulation algorithm (SSA): great for large deviations such as extinction
what is stochastic simulation algorithm (SSA)?
SSA considers stochasticity both in terms of demography and environmental; uses equations that describe how probability density of a system evolves in time
compare homogeneous x heterogeneous demographies exposed to irregular temperature fluctuations
Significant differences when comparing homogeneous x heterogeneous populations. In irregular environments, heterogeneous demography populations initially decrease, but later increase in comparison with homogeneous populations (this corresponds to comparing performance in constant x random fluctuations)
compare homogeneous x heterogeneous demographies exposed to sine wave fluctuations
In sine wave fluctuations, demographic events happen earlier in heterogeneous populations, resulting in higher pop. sizes than homogeneous ones
is it important to consider the heterogeneity of a population when modelling it?
yes, because studies modelling both exponential and logistic growth found big differences between the population sizes of homogeneous versus heterogeneous populations in a changing environment (both regularly and irregularly changing)
why is it important to consider the heterogeneity of a population when modelling it?
1) there is an established dependence between demographic traits and environmental conditions
2) climate change is increasing environmental variability
vital rates vary in terms of how they affect population growth rates: what is the main management implication of this?
management actions targeted at different vital rates will result in different population growth
what helps us to identify vital rates more likely to have a greater impact on population growth?
sensitivity analyses
how can we calculate fitness of invidivuals and populations?
based on lambda (growth rates)
what kind of population modelling analysis can be used to investigate natural selection?
sensitivity tools used for environmental management can be used for interpreting natural selection, as each vital rate can affect fitness to different extents
why identifying important vital rates is just first step when managing populations?
some vital rates are not manageable (constrained by nature: animal size; unethical: killing endangered species to improve survival of another; too expensive); we need to find optimal changes; we need to define if change is related to mean/variation in vital rates or to changing stage structure directly
why do particular dynamics of different ages, sexes, stages matter for predictions?
because changes in one stage can cause cascade effects on population as a whole
which biological characteristics can be used to define each population stage in a given species?
stage = age: humans and some other species
stage = recognizable morphology, size, behavior, etc.: most species
what does the 1st row of a matrix-projection model represent?
reproduction from each stage to next time step (eg., year) (reprod. contribution to next time step)
what does the diagonal row of a matrix-projection model represent?
proportion of invididuals surviving and keeping in the same stage next year
what do the rows and columns of a matrix-projection represent?
each stage
what does the subdiagonal of a matrix-projection represent?
proportion of individuals surviving and moving to the next stage
what does the sum of a matrix-projection column represent? (excluding 1st row)
total proportion of surviving individuals in a given stage in a given time step
what does a2,1 mean in a matrix-projection column?
proportion of individuals surviving the first time step and going to the next stage (eg., juvenile) in the next time step
what does a2,2 mean in a matrix-projection column?
proportion of individuals remaining in the same time step
which row in a matrix-projection allows for calculating stage-specific fecundity?
first row
what is fecundity in the context of stage-based population models?
average number of offspring produced by individuals in a given year (or other time step)
how can reproductive contribution to next time step be calculated?
based on fecundity (m) and survival (P)
does the timing of data collection matter for creating matrix-projections?
yes, because of the age of youngest individuals. If data was collected before new birth pulse, youngest individuals will have already developed; if it was collected collected after birth pulse, youngest individuals will be newborns
what is a vector in a matrix-projection population model?
a single column matrix with stage size for each year
how can we project future population sizes based on a matrix-projection and a vector?
we can obtain a vector (single column matrix with stage size for each year) and multiply it by the matrix-projection
how can we get the total population number per time step using a matrix-projection and a vector?
if we sum the rows of the vector resulting from the multiplication of the matrix-projection and the vector for the time step we get the total population number for the time step
how can we get the population growth rate per time step using a matrix-projection and a vector?
we divide the total population number in this time step by the total of the previous time step
what is a transient dynamic?
population dynamics following a disturbance that modifies the stable stage of this population (eg., harvest, translocations). When calculating the dynamics for this population for initial time steps and considering a constant environment, we see a fluctuation in the lambdas/total population size as time passes by which is solely due to impact of initial age distribution
what is stable stage distribution (SSD) / asymptotic lambda?
as we continue to calculate population growth across years, all stage-structured populations will eventually converge to a stable state (constant population growth rate; constant proportions of individuals in each stage); usually 20 time steps for vertebrates
asymptotic lambda growth rate remains the same across time in this stage
what is population inertia in a transient dynamics context?
caused by the lag time between initial stage structure and asymptotic vital rates & by the stage structure distribution (unequal for each structure): this causes population to grow a way smaller or bigger than if population were in asymptotic growth. Ex. “bubble” of reproductive females can lead to enormous growth for several years even if fertility is decreasing (population momentum)
what is reproductive value?
strength of each stage in terms of how much it contributes to future population growth: future population size/future population size of first age class -> allows understanding effect of inertia from transient dynamics on abundance
how can population inertia be calculated?
1- define how many years forward (eg., 14)
2- find SSD population: 100 (initial) x 1.34 (lambda) ^14 = 6018
3- find all adults: 684,102/6018 = inertia = 114 (population will be 114 times higher)
4- find all pre-juveniles: 2223/6018 = inertia = 0.37
describe 3 advantages of matrix-projection methods for wildlife population management
1- a set of vital rates represented as a projection matrix are powerful tools to predict proportion of individuals in each age group
2- this method allows for estimating uncertainty scenarios, eg., newly introduced population can be very erratic in terms of population growth if initial composition of population is far from SSD (stable stage distribution)
3- reproductive value can be used to understand the effect of removing individuals of a certain age (eg., harvest)
what is the importance of sensitivities and elasticities?
they show differences in the absolute (sensitivities) or relative (elasticities) IMPACT of each of the vital rates (survival or reproduction) to population growth rates
what are sensitivities/elasticities?
reproductive value and lambda SSD (stable stage distribution) are combined to assess how very small changes in different vital rates affect lambda SSD, while all other elements are held constant
what is the difference between sensitivity and elasticity?
elasticity rescales sensitivity to include the magnitude of a given vital rate (elasticities are proportional sensitivities)
give an example of how elasticity can be used to guide management of a long-lived species
long-lived species usually have higher elasticity for survival than for reproduction. Thus, following big changes in population growth, we should go and look first at adult survival
how can we add environmental stochasticity to matrix-projection age-structured models?
each time step has a different matrix with each element (vital rate) randomly chosen from distribution mean/variance (uniform or with central tendency)
when can demographic stochasticity can make a big difference?
small populations
what are main effects of stochasticity (variability) on stage-structured population dynamics?
variation in vital rates (especially ones with high sensitivities) reduces stochastic growth rates
OR
covariation (+ or -) in vital rates may either increase or reduce growth rates (DEMOGRAPHIC RATE COVARIATION)
sensitivities/elasticities by themselves do not explain extent of vital rates changes naturally/under management, so what do we need to do?
we need to analyze sensitivities/elasticities together with variation, because if a vital rate with small sensitivity varies a lot (eg., juvenile stage survival is more susceptible to predation/weather stress), it can end up having more impact in population growth than a rate with high elasticity (eg., adult survival) which does not vary much (this cannot be assessed with a mean matrix): LTREs (life table response experiments = considers vital rates variations)
what is negative frequency-dependent growth?
it can occur when intraspecific competition more strongly regulates a species’ growth rate than does interspecific competition (Adler et al. 2007).
what is fluctuation-independent stabilization in coexistence?
2 species coexist in both static and fluctuating environments (eg., negative frequency-dependent growth in 2 duckweed species)
when does a low-density growth advantage happen in a given species?
negative frequency-dependent growth (intraspecific)
what is the temporal storage effect?
when coexistence is facilitated by differential responses to a fluctuating environment. Competitive disadvantage an invading
species presents in static environments is not seen in a fluctuating environment.
what is stabilization (Armitage paper)?
In static temperatures, one species presents fitness disadvantages, but coexistence can still happen if negative frequency-dependent growth occurs. This negative frequency-dependent growth occurs due to interspecific niche differences. In fluctuating environments, stabilization can occur by a combination of buffered population
growth and differential responses to the environment
what are the main Armitage’s findings in terms of duckweed coexistence?
growth and competition did not differ in constant vs. fluctuating environments, as populations are stabilized (buffered) by negative frequency dependent growth (subtle niche differences). Reaction norms, dormancy, temporal storage (difference in competitive performance in constant x fluctuating env) effects are all weak (did not promote coexistence) (all are obvious differences between species)
what are the main findings of the mat density paper?
first order growth rate (r) depends on mat density (the more duckweeds, the less they grow)
what are the differences between dry biomass and growth rate in terms of mat density?
growth rate and mat density are highly inversely correlated (linear relationship); dry biomass grows together with mat density up to a point, after which it decreases
what is the relationship between nutrient removal rates (N and P) and initial mat density?
nutrient removal rates (N and P) vary depending on initial mat density (bell curve shape); at optimal initial densities, algae growth is inhibited
what are the 2 growth scenarios for duckweed? (mat density study)
free surface: growth is independent of mat density and dependent on environmental parameters
high surface densities: mat density limits or stops growth
why was a 2nd order equation developed for duckweed mats?
second order equation was developed to account for limitation given by mat density
how were experimental conditions kept during the duckweed mat experiment?
all constant, optimal values (nutrient, temperature, light intensity, photoperiod)
what controls microalgae growth?
duckweed mats prevent light from passing through
what is the population growth model more similar to the duckweed mat model?
is in accordance with logistic models (mat density = carrying capacity)
what kind of external factors are considered by the duckweed mat model?
in addition to key abiotic factors (light, nutrients, etc.) model also includes physical constraints (wind speed, water flow speed, rain)
considers harvest frequency
describe some parameters evaluated in the mat density models
optimal mat density, biomass yield, N and P removal rates,
under controlled medium eutrophication and physical con-
ditions
describe some variables used in the mat density models
harvesting frequency, limit mat density, intrinsic growth rate and N or
P content
what is the applicability of mat density models?
design and management of duckweed-based wastewater treatment facilities
what does the dominant eigenvalue represent in a projection matrix?
the lambda ssd (stable stage distribution), or asymptotic growth rate
it is a scalar which summarizes the effect (growth or shrinkage) of a given matrix
what does the right eigenvector represent in a projection matrix?
the ssd (stabe stage distribution)
the direction of growth or shrinkage of a population
what does the left eigenvector represent in a projection matrix?
vector of reproductive values representing the strength of each stage in terms of how much it contributes to future population growth
describe 2 characteristics of stable stage distribution (in terms of rate of growth and 2 scenarios with different starting distribution of individuals)
constant rate of growth for a given population
for 2 scenarios with different starting distribution of individuals, after reaching asymptotic growth, distribution of individuals becomes the same in both scenarios
what happens if we start projecting population dynamics using a stable stage distribution?
asymptotic growth will happen since the beginning, with the same lambda SSD from scratch
how are the values in a projection matrix calculated?
based on probabilities related to:
birth rates
surviving and staying or moving to next stage
and reproducing
