• N t+1
= N t
+ B – D + I – E
• ΔN = B – D + I – E
For a closed population
• ΔN = B – D

• dN/dt = B – D
• B = bN ; D = dN
(b and d are instanteous birth and death rates)
• dN/dT = (b-d)N
• dN/dt = rN
• N t
= N o e rt
1.1
1.2

Influence of r on population growth

• N t
= 2 N o
• 2N o
= N
• 2 = e r(td) o e r(td)
• ln(2) = r(td)
(td = doubling time)
• td = ln(2) / r 1.3

• No I or E
• Constant b and d (no variance)
• No genetic structure (all are equal)
• No age or size structure (all are equal)
• Continuous growth with no time lags

• N t+1
(r d
= N t
+ r d
N t
= discrete growth factor)
• N t+1
• N t+1
= N
= t
(1+r
λ N t d
)
• N
2
= λ N
1
= λ (λ N o
) = λ 2 N o
• N t
= λ t N o
1.4

• e r = λ if one lets the time step approach 0
• r = ln(λ)
• r > 0 ↔ λ > 1
• r = 0 ↔ λ = 1
• r < 0 ↔ 0 < λ < 1

• N t
= N o e rt ; where N t and r are means
σ r
2 > 2r leads to extinction

• P(birth) = b / (b+d)
• P(death) = d / (b+d)
• Nt = N o e rt (where N and r are averages)
• P(extinction) = (d/b)^N o

Elementary Postulates
• Every living organism has arisen from at least one parent of the same kind.
• In a finite space there is an upper limit to the number of finite beings that can occupy or utilize that space.

Think about a complex model approximated by many terms in a potentially infinite series.
Then consider how many of these terms are needed for the simplest acceptable model.
dN/dt = a + bN + cN 2 + dN 3 + ....
From parenthood postulate, N = 0 ==> dN/dt
= 0, therefore a = 0.
Simplest model ===> dN/dt = bN, (or rN, where r is the intrinsic rate of increase.)

Logistic Growth
There has to be a limit. Postulate 2.
Therefore add a second parameter to equation.
dN/dt = rN + cN 2 define c = -r/K dN/dt = rN ((K-N)/K)

• dN/dT = rN (1-N/K) or rN / ((K-N) / K)
• Nt = K/ (1+((K-N o
)/N o
)e -rt )

Further Refinements of the theory
Third term to equation?
More realism? Symmetry?
No reason why the curve has to be a symmetric curve with maximal growth at N = K/2.

What if the population is too small?
Is r still high under these conditions?
• Need to find each other to mate
• Need to keep up genetic diversity
• Need for various social systems to work

Examples of small population problems
Whales,
Heath hens,
Bachmann's warbler dN/dt = rN[(K-N)/K][(N-m)/N]

Instantaneous response is not realistic
Need to introduce time lags into the system dN/dt = rN t
[(K-N t-T
)/K]

Three time lag types
Monotonic increase of decrease: 0 < rT < e -1
Oscillations damped: e -1 < rT <

/2
Limit cycle: rT >

/2

Discrete growth with lags

1. N t+1
= N t exp[r(1-N t
/K)]
2. N t+1
= N t
[1+r(1-N t
/K)]
May, 1974. Science

Logistic growth with difference equations, showing behavior ranging from single stable point to chaos
(1) N t+1
= N t exp[r(1-N t
/K)]
(2) N t+1
= N t
[1+r(1-N t
/K)]

Added Assumptions
• Constant carrying capacity
• Linear density dependence