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Fig. 3.3. Increase in Monterey pine (P. radiata) in a eucalypt dry sclerophyll forest in Australia. Initially pine recruitment occurred from seed imported from an adjacent pine plantation. After 1980, recruitment rate increased even though the pine plantation was cut because pines established in the eucalypt forest were becoming mature and producing seed (redrawn from Burdon and Chilvers, 1994).

This is the logistic growth-curve equation which incorporates limits to population growth over time. When population density (N) is less than K, the term (K-N)/N will be positive and population growth will be positive. As the value of N approaches K, the rate of growth decreases until N=K when the rate of population growth (dN/dt) becomes zero. The population size is stable because births equals deaths at this time.

There are three parts to the logistic growth curve (Fig. 3.2b). Initially, population size increases at an exponential rate. The maximum rate of growth occurs at half the value of K. Beyond this, the rate of population increase slows down but is still positive. This occurs because not all individuals will be affected by limiting resources at the same time because of differences in size, age, health and reproductive status. Over time, the proportion of individuals affected by limiting resources will increase and this causes the curve to level off at K.

### Real population growth curves

The exponential and logistic growth curves are idealized mathematical descriptions of how population size will change over time. They provide a conceptual framework on which to base more complex approaches to population growth. In real situations, population growth is more variable over time (Fig. 3.4). There are a number of reasons why population size fluctuates over time. We will address a few here and you will see other examples in the rest of this text.

• The logistic growth model assumes that the environment is stable over time and therefore K remains stable. This is unrealistic because the abiotic environment is naturally variable: temperature, nutrients, water and light change over time. Even small changes in one factor can affect the number of individuals the environment can support.

• There is random variation in birth and death rates. This is termed demographic stochasticity. An occasional low birth rate or high death rate can cause the population to become extinct.

(a) Purple loosestrife in muck habitat

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Fig. 3.4. Examples of plant population changes over time showing: (a), (b), (c) the number of individuals and shoots of purple loosestrife (L. salicaria) in three habitat types (Falrnska, 1991), and (d) the mean number (number per/5 m2) of ramets and genets of blue grama grass (Bouteloua gracilis) (Fair et al., 1999).

• The logistic and exponential growth curves assume that populations are independent of other populations. Populations, however, interact (through competition, herbivory) and this causes population size to fluctuate. Population interactions are addressed in Chapters 8 and 9.

Effects of Migration (Immigration and Emigration) on Population Size

Sometimes it may be possible to ignore the effects of immigration and emigration (migration) by assuming that they are equal, or that their effect on population size is negligible. However, many will be dependent on the immigration of individuals from other populations. A population with fewer births than deaths will remain viable only when supported by seeds imported from other populations.

Migration demographically links populations. Determining whether migration is an important demographic process has two problems. First, the concept of migration assumes that there are specific boundaries over which individuals (seeds) move. In human populations we have political boundaries, so we can keep track of the movement of (most) individuals. As discussed in Chapter 2, plant population boundaries are rarely discrete. Second, even if 'real' boundaries do exist, tracking the movement of individuals can be challenging. Therefore, it is difficult to establish if migration is occurring.

Migration among populations: creating metapopulations

Traditionally, populations have been described as a collection of individuals that are capable of interbreeding. In reality, most populations are scattered and clustered into smaller subgroups. This clustering may be a random process but it usually reflects the heterogeneity of the landscape, i.e. there are a limited number of areas where individuals of various species can live and these individuals cluster in amenable habitats. When populations become divided into clusters, we can say that each cluster becomes spatially isolated from each other. If spatially isolated populations interact through migration (e.g. of seed) or distant pollination, then the aggregate of interacting populations is called a 'metapopulation'. The implication of using the term 'metapopulation' is that interactions among populations are not always common but they do occur.

Each population within a metapopulation will likely be genetically distinct because each is adapted to local environmental conditions. Although individuals within a population will mostly mate with individuals from their own population, metapopulation dynamics will introduce some genetic material from surrounding populations. Since the continued existence of a population is determined mainly by whether there is enough local genetic variation to withstand environmental change (including diseases, herbivory, drought) and ensure births exceed deaths, metapopulation dynamics may prevent the extinction of local populations. For example, immigrants (or at least their genetic material via pollen) from other populations can help maintain a population that otherwise would become extirpated (locally extinct) because it is not genetically suited to changes in its environment (e.g. decreasing light levels). Populations that are maintained only

Fig. 3.5. Metapopulation dynamics: population patches may be a source (bold) or a sink for seeds (or other propagules).

through immigration from other (source) populations are called 'sink' populations (Pulliam, 1988) (Fig. 3.5). In weedy white campion (Silene alba), for example, isolated populations survive only because new genetic material arrives via immigration from surrounding populations - in this case, the immigrant genetic material is delivered via pollen (Richards, 2000; see Chapter 4 on pollination).

Perhaps the most important aspect of the metapopulation concept is the implication for conservation. Because populations may contain relatively few individuals, be restricted to a small area or have low genetic variation, they are subject to local extinction. However, the metapopulation is usually persistent because local adaptations in populations increase the total amount of genetic variation. If the landscape-scale environment changes suddenly, the chances are good that at least one population has the genes needed to allow for recolonization of habitats vacated by local extinctions. This means that should a disease or a drought strike, then some of the populations will survive. Over time, this means that the local habitats that populations occupy often are 'emptied' and recolonized many times. Therefore while local populations may go extinct and the habitats emptied, the metapopulation of a species will continue. This has become important in understanding how to conserve species. It is possible

e) Random

Fig. 3.6. Theoretical age structure distribution used to assess population trends. The x-axis is the age class and the y-axis is the tree density (redrawn from Whipple and Dix, 1979).

that a large contiguous reserve that does not allow for spatial isolation, local adaptation, and development of a metapopulation can actually hasten extinction of a species, as it is vulnerable to sudden environmental change (Beeby, 1994; Hanski and Gilpin, 1997; Schwartz, 1997; Honnay et al., 1999; Etienne and Heesterbeek, 2000).

### Population Structure

Populations are characterized based on the age, size, appearance or genetic structure of individuals. In fact, population structure could be based on any characteristic that is variable within a population. Population structure is not a static feature of a population because individuals age, grow, reproduce and die at different rates depending on their individual characteristics and their environment. In this chapter, we focus on age, size and developmental stage structure of populations.

### Age structure

The distribution of ages within a population can be characteristic of the species itself, or it can reflect the 'health' of the population, or the environment inhabited by the population. In a 'healthy' population, younger individuals will outnumber older individuals because a proportion of young individuals will die before they reach maturity. Whipple and Dix (1979) proposed five age-class distributions to explain population trends of trees (Fig. 3.6). The 'inverse-J' curve shows a population with many more juveniles than adults; this population is likely to be relatively constant or increasing. The 'bimodal' distribution is a result of pulse recruitment (addition of new individuals) where periods of lower recruitment are followed by periods of higher recruitment. This population will likely be stable or increase as long as recruitment pulses are frequent enough to replace dying individuals. A 'decreasing' population distribution means the population is not replacing itself because recruitment is not high enough to replace those that are dying. If recruitment is zero the distribution will become 'unimodal' as the population ages and no young individuals are added. Although individuals are present, the population will become extinct unless increased reproduction occurs. Finally, a random distribution is typical of a population in a marginal habitat, or one that is responding to disturbance. Populations that have recently invaded a site are also likely to exhibit this distribution (Luken, 1990).

Age structures can be difficult to interpret because they do not always fit the theoretical distributions described above, nor are they consistent over time. Montana populations of spotted knapweed (Centaurea maculosa) tended to have inverse-J distributions in 1984, but in 1985 the distribution decreased (Fig. 3.7). This occurred following a severe drought in 1984, when young individuals experienced higher mortality than older individuals (Boggs and Story, 1987). While overall population density decreased by 40% between 1984 and 1985, the density of younger individuals (years 2 and 3) was reduced by 83%. This resulted in a change of age structure from one year to the next. The observed structure of a population is the result of abiotic and biotic forces encountered by previous generations of a population. It is important for scientists tracking changes in population density to be aware of age structure, because future changes in abundance depend very much on the current age distribution. As seen in spotted knapweed, harsh conditions may differentially affect age groups causing demographic changes.

There are complications, however, associated with characterizing populations based solely on age structure data. First, seeds that are persistent in the soil (seed bank) are seldom accounted for when assessing age structure of a population. The seed bank represents potential individuals that replenish the population when no new seeds are produced. Therefore, a population with no apparent seed production ('unimodal') may increase again via the seed bank rather than through renewed seed production. Second, not all plant species can be aged accurately

Fig. 3.7. Population age structure of spotted knapweeed at five sites in Montana (with individuals <1 year removed) (adapted and redrawn from data in Boggs and Story, 1987).

Age class (years)

Fig. 3.7. Population age structure of spotted knapweeed at five sites in Montana (with individuals <1 year removed) (adapted and redrawn from data in Boggs and Story, 1987).

and so age structure data may be suspect. Woody species (most trees and some shrubs) are easier to age than herbaceous species because they often produce annual growth rings which can be counted; however, not all woody plants produce annual rings, and some produce more than one ring in a year. Species producing multiple main stems will also be difficult to age. Some woody species can be aged by counting morphological features such as bud scars. Annual rings in the roots of some herbaceous perennials can also be used (Boggs and Story, 1987; Dietz and Ullman, 1998).

A third problem with using age structure data to characterize populations is that age may not be biologically relevant to population processes such as reproduction, growth or death (Werner, 1975). Two genetically identical individuals of the same age may differ physically depending on their environment and this will influence when they reproduce, the number of offspring they produce and when they die.

### Size structure

Most populations will tend to have fewer large individuals and many smaller ones. However, larger individuals can have a disproportionate effect on the rest of the population because they tend to live longer and produce more offspring than smaller individuals of the same age (Leverich and Levin, 1979). Larger individuals can also directly affect smaller individuals through shading. Plant size is a measure of the success of an individual because the larger individuals have acquired more resources than smaller individuals. For this reason, it is often more useful to structure populations by size rather than age. Furthermore, size may be a better predictor of an event (e.g. reproduction or death) than age (Werner, 1975; Werner and Caswell, 1977; Gross, 1981). Werner (1975) found that rosette size of teasel (Dipsacus fullonum) was a better predictor than age of whether a plant remained a vegetative rosette, flowered or died. For example,

Fig. 3.8. Annual survival and fertility (number of seeds per individual) of prayer plant (Calathea ovandensis). Individuals were classified into five stage classes (seed, seedling, juvenile, pre-reproductive and reproductive) with four size classes of reproductives (small, medium, large and extra large) (redrawn from data in Horvitz and Schemske, 1995).

Fig. 3.8. Annual survival and fertility (number of seeds per individual) of prayer plant (Calathea ovandensis). Individuals were classified into five stage classes (seed, seedling, juvenile, pre-reproductive and reproductive) with four size classes of reproductives (small, medium, large and extra large) (redrawn from data in Horvitz and Schemske, 1995).

rosettes attaining 30 cm in diameter had an 80% chance of flowering. Still, size is not a perfect predictor of life cycle events. An example of this is when small, repressed agricultural weeds flower even when they are tiny compared with their neighbours.

The simplest way to measure 'size' is to measure some visible aspect of growth such as plant height, diameter (e.g. of the stem), or number or size of leaves. Biomass is a more exact measure of size because it is a more direct measure of acquired resources, but biomass measurements require harvesting, drying and weighing the plant, and is a destructive sampling method.

A strong linear correlation between size and age rarely exists for many reasons. Some species of trees (e.g. sugar maple, Acer sac-charum) remain as slow-growing or suppressed individuals for decades until a canopy gap appears, after which they grow rapidly (Canham, 1985). Alternatively, plants may grow rapidly during the early life stages until they reach a maximum size and then divert resources to reproduction and maintenance rather than growth. Size structure also develops in shorter-lived species. In jewelweed (Impatiens capensis), size structure developed because larger individuals grew faster and had a lower risk of death than smaller ones (Schmitt et al., 1987). One should never assume that age and size are correlated until the relationship has been tested.

### Phenology

A plant's phenology (stage of development) can be used in conjunction with or instead of plant age and size to examine population structure (Sharitz and McCormick, 1973; Werner and Caswell, 1977; Gatsuk et al., 1980; Horvitz and Schemske, 1995; Deen et al., 2001). This measure may be more biologically meaningful than age or size alone because an individual's phenological stage may be more linked to its likelihood of survival or reproduction. Horvitz and Schemske (1995) showed that the annual

 Table 3.2. Life table of Drummond phlox (P. drummondii) (adapted from Leverich and Levin, 1979). Age at start of interval Length No. surviving No. dying Mean mortality (days) interval on day x Survivorship during interval rate/day x (days) nx lx dx mx 0 63 996 1.00 328 0.0052 63 61 668 0.67 373 0.0092 124 60 295 0.30 105 0.0059 184 31 190 0.19 14 0.0024 215 16 176 0.18 2 0.0007 231 16 174 0.17 1 0.0004 247 17 173 0.17 1 0.0003 264 7 172 0.17 2 0.0017 271 7 170 0.17 3 0.0025 278 7 167 0.17 2 0.0017 285 7 165 0.17 6 0.0052 292 7 159 0.16 1 0.0009 299 7 158 0.16 4 0.0036 306 7 154 0.15 3 0.0028 31 3 7 151 0.15 4 0.0038 320 7 147 0.15 11 0.0107 327 7 136 0.14 31 0.0325 334 7 105 0.11 31 0.0422 341 7 74 0.07 52 0.1004 348 7 22 0.02 22 0.1428 355 7 0 0 - -

survival and fertility of the prayer plant (Calathea ovandensis), varied depending on the individual's phenological stage (Fig. 3.8). Seedlings had less than 10% survival, seeds and juveniles had moderate survival while other stage classes had over 90% survival. Reproductive individuals produced different numbers of seeds per plant depending on their size.

### Illustrating population structured data

Data on age structure can be tabulated into life tables (Table 3.2). These tables summarize age-specific survival, mortality and reproductive rates. Survival data is used to construct survivorship curves that display the proportion of individuals surviving to the beginning of each age class (Fig. 3.9). Survivorship curves are easier to interpret when presented on a log scale because they show constant mortality rate as a straight line. A steeper slope indicates a higher mortality rate.

Pearl and Miner (1935) presented three general survivorship curves (Fig. 3.10). These model curves are often referred to as Deevey curves after Deevey (1947). Type I is typical of species, such as some human populations, with low early mortality, and high mortality later in the life span. A Type II curve shows a constant mortality rate throughout the life span. Some birds have this type of survivorship curve. A Type III curve has high early mortality that decreases later in the life span. This is typical of many plant species where seedling mortality is very high (e.g. agricultural weeds). When presented on an arithmetic scale, the curves appear different. The methodology for collecting and calculating survivorship data will be explained in Chapter 10.

Why does population structure matter?

Interpreting population structure can be difficult and time consuming. Why, then, do we do it? Why not simply calculate population means (e.g. mean age or height) and use these simple numbers to describe a popula-

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