Population Dynamics and
Growth

Population Dynamics
• Population distribution and abundance change through time – not static features, but ones that are in constant flux

Population Dynamics
• Patterns of population distribution and abundance arise through the dynamic balance of two types of processes:
• processes that ADD individuals to populations and those that REMOVE individuals from populations

Population Dynamics
+ Immigration & births – add individuals to populations
Emigration & deaths – remove individuals from populations
Emigration Immigration

• Many species have a stage in their life cycle that is specialized for long-distance dispersal
• winged seeds, fruits
• pelagic larvae of many sessile marine invertebrates and nearshore fish
• spores
• winged stage in some otherwise wingless insects (aphids, termites)

• movement of individuals to establish populations in new areas (range expansion)

• Range expansions often track changing climatic conditions

Changing vegetation distribution in Santa Rosa Canyon from 1977 to 2006
–2007
Kelly A. E., Goulden M. L. PNAS 2008;105:11823-11826
©2008 by National Academy of Sciences

• movement of individuals within the bounds of an established population range

Great white sharks

• movement of individuals between spatially isolated subpopulations: Metapopulation dynamics

• individuals in small subpopulations are more likely to disperse than those in large subpopulations
• females disperse more frequently than males
• individuals are more likely to disperse to a subpopulation that is larger than the one they left

Population Dynamics
+ Immigration & births – add individuals to populations
Emigration & deaths – remove individuals from populations

Population Dynamics - Survivorship
• Survival patterns vary greatly between different species
• Life tables are a tool used to document survivorship and deaths (mortality) in a population

Population Dynamics - Survivorship
• Cohort life table : a record of survivorship is kept for a large group of individuals that were born at about the same time – the cohort is studied until every individual has died.
• This approach is the most reliable, but also the most difficult to implement in the field
(particularly for long-lived or mobile organisms!)

Population Dynamics - Survivorship
• Static life table : the age at death is recorded for a large number of individuals (that weren’t necessarily born at the same time).
• This can be achieved through tagging individuals when born, then noting age at death…or by estimating the age of dead individuals (e.g., by counting growth rings)

Population Dynamics - Survivorship
• Age distribution : the proportion of individuals in progressively older age classes within a population is used to estimate survival (assumes that difference in number of individuals from one age class to the next is due to mortality)
• This method assumes that a population is neither growing nor decreasing, and that no dispersal is taking place – since these assumptions are almost never met in real populations, this method is the least accurate.

Survivorship in Dall sheep
• Static life table was constructed from age at death determined from skulls (horn growth rings and tooth wear)

Dall sheep – Survivorship curve

Constant rates of survival: Birds

High juvenile mortality
• Some organisms produce very large numbers of young with high mortality rates
• aquatic species with pelagic eggs/larvae
• plants that produce large quantities of seed

High juvenile mortality

Three Types of Survivorship Curves
• Type I survivorship curve: a relatively high survivorship rate among young and middle aged individuals, high rate of mortality for older individuals
• Type II survivorship curve: constant rates of survival throughout life
• Type III survivorship curve: extremely high rate of juvenile mortality followed by relatively high rate of survival

Three Types of Survivorship Curves

• The age distribution of a population (numbers of individuals in different age classes) reflects its recent history of survival and growth, as well as its potential for future growth
• Age distributions can be used to identify periods of successful reproduction, periods of high and low mortality, and whether or not the population is replacing itself as older individuals die.

• In a stable population, the age distribution is skewed towards younger individuals, meaning that the older individuals are being replaced as they die
• In a declining population, the age distribution is dominated by older individuals, indicating that young individuals are not being produced or are not surviving to replace older individuals as they die

Stable and Declining Populations

Rates of Population Change
• In addition to survival, the rate at which new individuals are being born into the population also has a direct influence on population density
• Birthrate = the number of offspring (live births, eggs, seeds, etc.) produced by a parent per unit time

Rates of Population Change
• Average number of births per female for each age class and the number of females in each age class: Fecundity Schedule
• A fecundity schedule can be combined with a life table to estimate several important population characteristics:
• net reproductive rate
• geometric rate of increase
• generation time
• per capita rate of increase

• Net Reproductive Rate, R
0
, indicates whether a population is growing (>1) or declining (<1)

• about half (0.507) of female turtles nest each year
• most that nest do so only once, but about 1/5 nesting females nest two times, so…
• nesting females produce 1.2 nests / year
• average clutch size is 3.17 eggs

• Generation Time, T , is the time from egg to egg, seed to seed, mother to offspring, etc.
• Per Capita Rate of Increase, r , is related to the generation time and net reproductive rate as:
r = ln R
0
/ T positive values indicate the population is increasing/stable, negative values indicate that the population is declining

For the mud turtle population, the Per Capita Rate of Increase, r =
r = ln R
0
/ T
r = ln 0.601 / 10.6
r = - 0.05
…the negative value here indicates that the population is declining (birth rate is lower than the death rate)

Population Growth
• The environment has a direct effect on population dynamics: the growth and decline of populations is influenced directly by the environment
• Depending on their reproductive biology and environmental conditions, populations can grow geometrically, exponentially, or logistically

• When the environment is favorable and resources are abundant, populations can grow at geometric or exponential rates
• Growth will occur slowly at first, then growth will accelerate to increasingly faster rates

Geometric Population Growth
• This pattern of population growth is displayed by organisms that have a pulsed pattern of reproduction – that is, they produce a single generation which is then completely replaced by the next generation (no overlap in generations)
• e.g., annual plants

Geometric Population Growth
• Because these populations grow in discrete pulses, successive generations differ in size by a constant ratio
• This ratio is λ , the geometric rate of increase

• λ is the ratio of the population size at two points in time:
λ = N t+1
N t
N t+1 is the size of the population at some later time
N t is the size of the population at some earlier time

λ = N t+1
/ N t
λ = R
0
N t+1
= (996)(2.4177) = 2408
Nt = 996
λ = 2408 / 996 = 2.4177

Geometric Population Growth
• For populations whose generations do not overlap, growth can be modeled by simply multiplying λ by the number of individuals in the previous generation:
• N
1
• N
2
• N
2
• N
2
= N
0
× λ
= N
= N
= N
1
×
0
×
0
×
λ
λ × λ
λ 2
N t
= N
0
λ t

Geometric Population Growth
• For many reasons, unlimited population growth cannot be maintained for very many generations, and λ will decrease
• insufficient space
• insufficient resources
• limited suitable habitat

Exponential Population Growth
• Organisms that have overlapping generations can have continuous population growth, and the geometric model is not appropriate
• Continuous population growth in an unlimited environment can be modeled as exponential population growth

• dN / dt = r max
N
• change in numbers of individuals with change in time = the maximum per capita rate of increase attained by a population under ideal conditions (where birth rate, death rate, and age structure are constant) times the number of individuals (N)
• r max is called the intrinsic rate of increase
• because the environment generally limits unchecked exponential growth in some way, r, the actual per capita rate of increase, is almost always less than r max

• The population size at any time, t, for a population growing exponentially can be calculated as:
N t
= N
0 e r max t
N t
= number of individuals at time t
N
0
= initial number of individuals
e = base of natural logarithms r max
= intrinsic rate of increase, in offspring per time interval
t = number of time intervals (in days, weeks, years, etc.)

Exponential Population Growth
• In reality, natural populations can grow exponentially, but usually only for a limited time before resources are depleted
• Exponential growth is most common in several circumstances:
• during the process of establishment in new environments
• during exploitation of transient, favorable conditions
• during the process of recovery from some form of exploitation

Exponential Population Growth

Exponential Population Growth

Exponential Population Growth

Logistic Population Growth
• As exponential population growth slows due to environmental constraints and eventually stops, a new growth model emerges – logistic population growth

Logistic Population Growth

Logistic Population Growth
• Carrying capacity, K , arises through the interplay of many possible factors:
• food
• parasitism
• disease
• wastes
• space

Logistic Population Growth
• model of logistic population growth is just a modification of the exponential growth model that incorporates an element that slows growth as the population size, N, reaches the carrying capacity, K
• dN / dt = r max
N (1 – N / K)
…so, as N gets bigger, N/K also gets bigger, and
1 – N/K approaches 0 (at which point, population growth is also 0).

• r, the actual per capita rate of increase is equal to
r = r max
– r max
(N/K)
r = birth rate – death rate, so when N is small, the birth rate greatly exceeds the death rate.
…as N increases, r becomes smaller, meaning that the birth rate is approaching the death rate
…when N = K and r becomes zero, the birth rate and death rate are also equal
…if N becomes greater than K, r will become negative, and the death rate will exceed the birth rate until N = K again.

birth and death rates
• Both biotic (diseases, predators, food…) and abiotic
(floods, earthquakes, cold, drought…) factors act to alter the birth and death rates in population
• These factors alter K, thereby influencing the birth and death rates in populations