**Lecture 29 (10-Apr-13)**

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As the first step in discussing population regulation, we will look
at the interplay between density-dependent (**DD**)
forces and density-independent (**DI**)
forces from a graphical perspective. The graphs will be somewhat
similar to the harvest graphs of Lecture
23. That is, they will have *N* on the *X*-axis and d*N*/d*t
*plus
linear functions on the *Y*-axis.

We will incorporate the **DD**
forces via logistic influences of a population's density-impact on
itself.
The **DD**
forces
produce humped d*N*/d*t *curves (look back at Fig. 6.1 of Lecture
6). "Good" **DD**
years have higher maxima (top of the hump) and larger *K* (how
far
the end of the hump goes out the *X*-axis, describing *N*).
"Bad" **DD**
years have lower maxima and left-shifted *K*. We will
incorporate
the **DI** forces via "harvest-like"
linear functions. Strong **DI** (more
animals taken out by a factor such as bad weather) means a steeper slope.

The "currency": we will measure the impact of **DI** and **DD**
on ** equilibrium population size**,

A graphical analysis of the interaction between density-dependence and density-independence:

** Motivation and system setup**:

Imagine that we have both some density-dependent
(**DD**) and density-independent (**DI**)
factors influencing the population of a species of interest. For concreteness,
say that the population is the Wyoming toad (*Bufo* *baxteri*).
Use the following characteristics and assumptions for modeling the system:

1) Let theNow, say we are monitoring two populations of toads -- a "northern" population at the north end of the Laramie Basin and a "southern" population at the south end. We get population fluctuations that look like the two graphs below.DDfactor be predator load (whoseper capitaeffectchangesas the toad population changes)

2) Let theDIfactor be rainfall (whoseper capitaeffectis constantregardless of the population size).

3) GoodDDyears have low predator loads, badDDyears have high predator loads;

4) GoodDIyears have high rainfall, badDIyears have low rainfall.

5) Assume that theDIvariation is as large, or larger, than theDDvariation.

6) Finally, assume predator load (DD) and rainfall (DI) vary independently (that is, they are not correlated).

In the northern population, rainfall gives us a pretty accurate indicator of population trends. Shortly after rainfall peaks we will tend to get population peaks. [Depending on the (unspecified) time scale, this might mean that the population responds with a one-year time lag to changes in rainfall -- this might be because we census "adult" toads and the high numbers this year represent good recruitment last year].Fig. 29.1.Population fluctuations of a toad's population size (N) and rainfall (in mm) over time (t) for the "northern" population. Black, dashed line: rainfall, with a mean of approximately 25 mm per year, ranging from approximately 10 to 40 mm. Over the same time period, the population fluctuated from approximately 50 to 90 individuals, with an approximate mean of 70. Note thatthe population "tracks" rainfall fairly closely-- with a slight lag.

The good correlation in the northern population contrasts with the lack of correlation shown in the graph below for the southern population.

Fig. 29.2.Population fluctuations of a toad population (N) and rainfall (in mm) over time (t) for the "southern" population. Black, dashed line: rainfall, with a mean of approximately 25 mm per year, ranging from approximately 10 to 40 mm (the rainfall is the same at both sites). Over the same time period the population fluctuated from approximately 65 to 75 individuals, with an approximate mean of 70. Note thatrainfall is poorly correlated with population trends.

I will use a revised version of one of the harvest models we examined earlier ( Lecture 23) to address one way of examining/modeling the patterns given in Figs. 29.1 and 29.2.

We will have humped curves, depicting **intrinsic density-dependent** (**DD**) factors
(effect varies with density: low growth at small *N*, to high growth at intermediate *N*, then back to low) via logistic growth.

We will have straight lines, depicting density-independent (**DI**) factors that affect population
growth (they affect an unchanging proportion of the population). We will allow the density-dependent factors to vary between "good" (higher
humped blue curves, with higher *K*) and "bad" (lower humped blue curves, with lower *K*).
The density-independent factors will vary between "strong" (steeper-sloped dashed straight lines) and "weak"
(shallower-sloped dashed lines).

Our approach will be to assess the difference between **equilibrium population size** under one combination of regulating forces versus some other combination. We will ask the question: "What makes the biggest difference: **DI** differences
(strong vs. weak) or **DD** differences (good vs. bad)?"

Let's contrast a case where the

Fig. 29.3Interaction between density-dependent (DD) effects on population growth (solid blue curves) and density-independent (DI) effects (red, dashed lines). The difference between good and bad years is large forDDeffects (differencesaminusbandcminusd, holding theDIeffects constant), compared to the difference between strong and weakDIeffects (aminuscandbminusd, holding theDDeffects constant). Note that pointsa, b, c, anddrepresentequilibrium population sizes -- the sizes at which addition (blueDDcurves of logistic growth) are balanced by subtraction (red, dashedDIcurves of density-independent removal). Four curve intersections means four equilibrium points.

Fig. 29.4.Interaction between density-dependent (DD) effects on population growth (solid blue curves) and density-independent (DI) effects (red, dashed lines). With the steeperDIlines, the relative importance ofDIandDDare reversed when assessing effects under variation. The difference between good and bad years is now large forDIeffects (aminuscandbminusd, holding theDDeffects constant) compared to those forDDeffects (differencesaminusbandcminusd, holding theDIeffects constant).

What are some major differences between the forces acting in Figs. 29.3 and 29.4?

1) In Fig. 29.3 the density-dependence is more dramatic (difference between

Kin good vs. bad years is greater)

2) In Fig. 29.3 the density-independence is weaker (both curves have lower slope than in Fig. 29.4)

3) In Fig. 29.4 we have decreased theDDgood-bad difference

4) In Fig. 29.4 we have increased the impact ofDIfactors (made the lines steeper).

If we use the same notation to label the points as *a, b, c*, and *d*, then **DD** dominates
when *b* > *c*, while **DI** dominates when *b* < *c*.

**Outcome: DD dominates in Fig. 29.3, while DI
dominates in Fig. 29.4.**

**Now we will turn to an equation-based analysis of the same issues.**

**Equation-based exploration of DI vs. DD as the main regulating force:**

NOTATION: for all the graphs and equations below, I will use the subscript "**G**" to refer to parameters
associated with **G**ood** DD** years, and "**B**"
to refer to parameters associated with **B**ad **DD**
years. I will use the subscript "**S**" to refer to **S**trong** DI**
impacts (more animals taken out, steeper slopes) and "**W**" to refer
to **W**eak **DI**
impacts (fewer animal taken out, shallower **DI**
slopes).

Base equations for the four curves (all of them are d*N*/d*t *type differential equations):

DD_{G}=rN(1-N/K_{G})Eqns 29.1DD_{B}=rN(1-N/K_{B})DI_{S}=R_{S}NDI_{W}=R_{W}N

where the first two are simply the logistic equation -- with a subscript on the K.
Look back at Fig. 6.1 to remind yourself that the graph of logistic dN/d
t against N is a humped curve. The
second two are linear (*Y*-intercept is 0, slope is *R*_{i}, where the i
subscript can be S strong effect, or
W, weak effect). Think of the *R*_{i} as a proportion of the animals
removed by the **DI**
factor (that **DI** factor could be harvest, weather mortality, or something else).
The four equations 29.1 produce the four blue or red curves of Figs. 29.3 and 29.4. Remember,
the **DD** are adding animals,
while the **DI** are removing animals. The combination of the humped curve for the
addition and the linear increase for the removal is very similar to
what we looked at with linear harvest (see Fig. 23.3).

The four curves lead to four values of N where **DD**
and **DI**
curves will intersect – **the equilibrium points**. We will use the symbol "&" to mean intersection.
**DI**_{S}&**DD**_{B}
, for example, will mean the intersection of the **DI**_{S }line and the **
DD**_{B} curve.
We can solve for N at those
intersection/**equilibrium** points by setting pairs of the equations in 29.1 equal to
each other. Those intersection N-values
are given by:

a=DI_{S }&DD_{B}=K_{B}[1-(R_{S}/r)]Eqns 29.2

b =DI_{S }&DD_{G}=K_{G}[1-(R_{S}/r)]

c =DI_{W }&DD_{B}=K_{B}[1-(R_{W}/r)]

d=DI_{W }&DD_{G}=K_{G}[1-(R_{W}/r)]

The "switcher" intersections arebandc. Their relative magnitude will determine whether DD or DI will dominate.

[Note thatais always less than, andbis always less thancd, but the order ofandbcan vary, as we will see in some of the graphs below]. Again, remember that each of these is ancequilibrium pointunder a given combination of conditions.

Whybandcmatter (usingaas the reference point):

DI dominates:If<bthencato(changingbDDwhile holdingDIconstant) is

shorter thanato(changingcDIwhile holdingDDconstant)

Small (ato) distance means that the effect of changingbDDis small --DIdominates.

Large (ato) distance means that the effect of changing DI is large --cDIdominates

DD dominates:If>bcthenato(changingbDDwhile holdingDIconstant) is

longer thanato(changingcDIwhile holdingDDconstant)

Largeatomeans that the effect of changingbDDis large --DDdominates.

Smallatomeans that the effect of changingcDIis small --DDdominates

Let's go back to the idea of using the difference between **DI** and **DD** intersections (the change in equilibrium values as we change conditions) to decide which one dominates.

What does the order of *b* and *c* mean again?

If *b* < *c* then *a*-*b* and *c*-*d* (changes in **DD** holding **DI** constant) are small relative to *a*-*c* and *b*-*d* (changes in **DI** holding **DD** constant). That is, the **change in the equilibrium population size** across **DI** changes is small relative to the change in equilibrium population size for **DD** changes. Compare the graphs of Figs. 30.1a and 30.1b as examples.

In the Fig. 29.5 case [small variance (

Fig. 29.5.DDdominates. Big {atob} and {ctod}. Dashed lines anda-b,c-d(etc.) "effect" notes are intended to provide context for the meaning of the order ofbandcto be compared with Fig. 29.6.

Fig. 29.6.DIdominates. Compare and contrast with Fig. 29.5. The horizontal red and blue bars below the graph illustrate theeffect on(measured along theNX-axis) of changes inDIorDDeffects (given by the two humped curves and the two slanting red solid or dashed lines). By increasing the difference between the solidDI_{S}and the dashedDI_{W}lines we have switched fromDDdominance toDIdominance.

In the Fig. 29.6 case [large variance(**DI**)] the effects on *N* are reversed.

Let's try two different approaches to answering questions about **DI** vs. **DD**.
The first will be a "numerical and word-problem" approach, the second will be graphical.
Both will share certain characteristics.

**Numerical, word problem approach**: Say I give you the following "data"

N*Intersect. 630 DI_{W}&DD_{B}495 DI_{S}&DD_{B}700 DI_{W}&DD_{G}550 DI_{S}&DD_{G}

How should you approach the problem? Our main interest is the effect (on *N** the expected net growth produced by **DD** - **DI**). What factors affect the effect? The **DD** and **DI** curves.

Effect of *X* holding *Y* constant vs. effect of *Y* holding *X* constant, should tell us which matters most. In this case, the *X* and *Y* are **DD** and **DI**. How do we evaluate the effect? We assess the effects as the **difference** between good and bad ,or weak and strong, holding the other factor constant.

In no particular order:

effect of **DI** while holding **DD** constant on **DD**_{B} curve: 630 - 495 = 135

effect of **DI** while holding **DD** constant on **DD**_{G} curve: 700 - 550 = 150

effect of **DD** while holding **DI** constant on **DI**_{S} line: 550 - 495 = 55

effect of **DD** while holding **DI** constant on **DI**_{W} line: 700 - 630 = 70

Verdict? The **DI** effects (both > 100) are considerably larger than the **DD** effects (both < 75), so **DI** dominates.

**Graphical approach**:

Say I provide you with the following graph and ask you to explain which factor dominates:

Again, we solve the problem by holding things constant (staying on a

Fig. 29.7. PossibleDIvs.DDregulation question.Question: Given the graph above, determine whetherDDorDIwill dominate, and explain your reasoning. [Answer is Fig. 29.8]

Fig. 29.8.Solution of the question graph in Fig. 29.7. Drop lines [the places where they hit theX-axis tell us the equilibrium population sizes] and an explanation of the horizontal bars connecting them are a sufficient explanation of the outcome. Here, the effects of changingDIconditions (red bars) is much less than the effect of changingDDconditions (blue bars), soDDdominates.

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