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Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. 3. The Population Dynamics of Extinction. This section is all about the fact that in order to conserve a species it may not be enough just to leave it alone and stop hunting it or destroying/polluting its habitat. There are various things that happen once the population’s size has been reduced seriously. The figure below shows the various mechanisms that are important at various sizes. Healthy Large and wide spre ad populati on *Habitat loss or degradation *Hunting *Habitat Fragmentation *Global catas trophes Vulnerable e.g. Heath Hen; Si ngle location or Blue Whale; wide spread low densi ty N. Elephant s eal. The above plus … *Environmental stochas ticity *Local catastrophes Critical Very small numbe rs or very low de nsi ty All the above plus … *Allee effects *Demographic stochas ticity Extinction Figure 3-1. Chart showing the major possible factors involved in the road to extinction, including those causing increased vulnerability to extinction in small populations. Page - 16 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. 3.1. Extinction in an Impoverished Environment. The most obvious way in which we can model the extinction of a species is when an organism is placed in a poor environment. This is one in which the number of deaths per unit time is greater than the number of births. Let N be the number of organisms in the population, let N0 be the initial number of organisms, t be time, and let B and D be births and deaths respectively per adult in the population. Then the population will develop according to the equation: N t +1 = (1 + B − D) N t , t=0,1,2,3…. …(3-1a) So the replacement coefficient is λ = 1+ B − D = 1+ r …(3-1b) The r constant is the growth rate. Clearly if D>B, then the replacement rate is less than unity (and r<1), and so the population is exponentially declining, as you saw in Fig. 2-2a. Unless action is taken, extinction will follow. This system is relatively elementary but it may arise in various natural situations. For example, we might introduce a colony, of size N0, into a patch of poor habitat. Alternatively, a population of size N0, at a given time may experience a change in the conditions prevailing in its habitat, going from that of Figure 1b from that of 1a. 3.2. Extinction through Allée Effects Sometimes populations are subject to additional strains simply from being very small through the Allée effect that leads to a reduced λ at low numbers. If such an effect is operating in a population, reducing the numbers of an organism may lead to a chain reaction resulting in extinction. An Allée effect can arise in a number of ways. • The community structure may collapse. The passenger pigeon is believed to have suffered extinction for this reason. • The population density (number per area) may be very low so animals may have difficulty locating each other to achieve a mating. This has been suggested (though unproven) in the case of the blue whale (Balaenoptera musculus). • Inbreeding depression may induce extra mortality. When a large population is suddenly reduced to a small size, many deleterious recessive genes are "exposed" by the increased inbreeding, leading to reduced fitness. However, not all inbred populations suffer from inbreeding depression. Models nicely describe the influence of the Allée effect. We can set up the model according to the Eq. (3-1) but with an extra mortality term that is small but gets large when the number of animals is small: δ Λ( N ) = 1 + r − …(3-2) N This means that exponential growth is modified by an extra factor: N t +1 = (1 + r − δ / N t ) ⋅ N t Page - 17 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. In the sheet I have given you, for the graphical method, I have already shown the replacement curve, with this effect. Use your graphical skills to see what happens for different starting populations, using the figure provided Fig 2.2(c). What happens? • Is there an equilibrium? • If the equilibrium exists what is its value? We can also find the equilibrium value from the equation above. It is: When there is an Allée effect operating the important consideration is that there is a minimum viable population, below which the population cannot survive and above which it can survive. 3.3. Environmental Variability and Catastrophes. These two mechanisms fall into the category of things that affect medium size populations. The effect can be sweeping as the case of the heath hen showed. Introducing environmental variability into a model requires that things like birthrate must vary with time also, not just due to the randomness inherent in births and deaths, but in a systematic way. Instead of staying at a value of K, the maximum value of the population will vary from generation to generation. Population Time Figure 3-2. Fluctuations about a carrying capacity due to environmental stochasticity in the replacement rate, and (lower curve) due to occasional epidemics of varying sizes. Notice the slow recovery in case of the lower curve. In such cases it is common to build a Logistic or Beverton-Holt equation, for example, with the parameters fluctuating. How can we do this? One possibility is actually to go out and measure both r and K for several years to see how much they vary. We then include this in the model and try to see how often the population goes extinct. Page - 18 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. I have included a file on the computer in EXCEL with which you can analyse the behaviour of such a model. Figure 3-3. Copy of the excel spreadsheet when you open it. You can use this to investigate the effects of environmental stochasticity on the growth rate and the carrying capacity, and of changing other parameters. You should try σr=1%, 5%, 10%, 30%, 98%and 60% with σK=0, and vice versa, for each of r=0.3 and r=0.03. Although this method helps us to understand the mechanism of extinction, it has limitations. The main problem is that the environmental variability is assumed to be constant in variance. However, we know that variance increases with time: more timeàmore variation. Because, as we take in more time, all sorts of things which seemed “stationary” on short timescales start to “kick in”. For example ice-age cycles are not noticeable on short timescales, but they dominate fluctuations on timescales of 10-thousand years to 100 thousand years. If we want to predict extinction on longer timescales we must include such effects in our models. Page - 19 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. 0.30 0.4 T T T 0.2 0.25 Rel. Temp. (Degrees C) 0 0.20 -0.2 StDev -0.4 0.15 -0.6 0.10 -0.8 -1 0.05 -1.2 1200 1400 1600 1800 2000 0.00 1 10 100 1000 Date T Figure 3-4. More time more variation for environmental stochasticity! On the left is a reconstruction of termperature in the Northern hemisphere over the last 750 years (heavy black line is the smoothed version). We find that the amount of variability (variance) over an interval of 100 years tends to be greater than that for 10 years. Changing weather patterns could be described as “environmental stochasticity” and epidemics might be described as “catastrophes”. However, both are essentially part of the same complex process that we might call “the environment”, since they are extrinsic to the population rather than intrinsic. The complexity of this is enormous, but we hope that some patterns may help to put some order in the modelling of the environment. 3.4. Fragmentation of Populations and Immigration. One of the characteristics of Man's current effect on the environment is an increasing level of habitat fragmentation. As cities, motorways and agricultural areas expand, those areas left to "Nature" contract and become fragmented. This has harmful effects: • Smaller population are more likely to be exposed to the harmful effects of demographic stochasticity (next section). (“Together we stand, divided we fall”). • Inbreeding is more serious problem in smaller populations. • Usually the borders of habitats suffer from various edge effects that effectively reduce the effective population in each patch. On the other hand, fragmentation may be a good thing. If there is plenty of distance between the patches but still reasonable migration, there will be a • built-in protection against epidemics • Better protection against geographically localised natural disasters. Page - 20 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. Figure 3-4. Two types of “metapopulation” geometry. Island archipelago (left) and mainland-island (right). Trying to quantify the relative merits of these options, in conservation biology, is called the SLOSS (single large or several small) problem. This problem continues to be the subject of research and it has not been solved entirely. In cases where there is reasonable immigration into the outlying patches from other populations is significant there can be restoration of populations. The outlying islands become part of a metapopulation, a fragmented population, linked by dispersal and recolonisation. If the island cannot survive by itself, it may still appear to do so, having a small population. Carry out the graphical procedure for Fig 2-2(d). Such a population is referred to as a sink population, and the population that supports it as a source population. In such studies, local extinction refers to the loss of a population in a single island or patch. Global extinction means that the population is lost from all islands or patches, so that there is no chance of recolonisation. Page - 21 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. 3. 5. Extinction by Demographic Stochasticity. When it happens, extinction usually involves “demographic stochasticity”. This means that although the external conditions causing extinction may be well-defined, the exact time is uncertain because the internal dynamics of individuals are chancy, since birth-rate, death-rate and sex ratio only exist as averages. For example, a population is truly doomed, though it may be “large” when all individuals are of the same sex, since there are no further offspring. There are three different cases to consider the effects of demographic stochasticity. 1. Declining Population. Fig 3-4a, b shows the results of a Monte-Carlo simulation model used to simulate the fate of 15 identical island populations that have been simultaneously subjected to a degradation of their environment. Shown in Fig 3-1 are average population (a) and number of surviving populations (b). Each population starts at 100 individuals. Although the exact time of extinction is not certain, there is clearly a characteristic time when the status of a population becomes critical, in this example it is when the population falls to a value of about 10. 2. A Colony That Fails to get Established. The other two panels in Fig 3-4 show the reverse process; colonisation by an organism of a favourable environment, subject to demographic stochasticity. Although the average population (c) increases exponentially, the fate of individual islands is chancy. Any colony may dribble along at very low numbers, before it “takes off” into exponential growth. This transition is the establishment of a colony. Over half the initial colonies (of 10 animals) in this model went extinct before they have a chance to get established. One of them petered out after 50 years. For this reason, remote islands, subject to such colonisation processes, often contain very different faunas and floras although conditions might be very similar. A species may be fortunate to establish itself on one island, yet fail to get established on another. A similar question, with the same mathematical model, is whether a mutant gene can invade a population of otherwise normal genes. In the early stages of colonisation, a population is very vulnerable to demographic extinction. For populations starting from a small number of individuals, there is a high probability that we might have all males or all females in the population, or that the few individuals in the population might suffer reproductive failure. Page - 22 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. (a) Deathrate>Birthrate (Exponential decline) (c) Deathrate>Birthrate (Exponential increase) -0.1129x y = 118.11e 50 50 40 40 Exponential y = 4.8313e 0.0643x Population size model Population size 30 30 20 20 10 Exponential 10 model 0 0 0 20 40 60 0 10 20 30 40 Time (years) Time (years) Extinctions begin when average populations falls below N~10 (b) 120 (d) 120 Number of Surviving Populations 100 100 Number of Surviving Populations 80 80 60 60 40 40 20 20 0 0 1 11 21 31 41 51 61 71 1 11 21 31 41 Time (years) Time (years) Figure 3-5. Extinctions due to demographic stochasticity in an ensemble of 20 identical populations with (left) negative growth-rate and (right) positive growth-rate. Page - 23 Document converted by PDFMoto freeware version Dr. J.M.Halley, Extinction Models; Part 3. What is the probability that a colony, beginning with N0 individuals in a favourable environment, gets established? This problem has been solved mathematically for simple population dynamics. Suppose that organisms breed (one offspring at a time) at any time with probability B per individual per unit time and die (one at a time) at any time with probability D per individual per unit time. Notice that the birth and death probabilities are independent of the number of individuals present on the island and that we don’t distinguish between males and females. Obviously, this is a fiction; however, the error is very small. The answer is given by the following formula: D N0 PE = , B > D B Initial colony size is N 0 …(3-3) =1 , B≤D This formula may also hold true from a population recovering from a serious population loss. So if we include demographic stochasticity, the population only survives with a probability, its survival is not certain. Nevertheless, even if the rate of births is much greater than the number of deaths, the chance of extinction is small. 3. Extinction of Established but Small Populations. Again because of demographic stochasticity, there is a constant fluctuation about the ceiling or carrying capacity. Most of these fluctuations are small, but occasionally there are large ones, big enough to cause extinction. Even for an established population, if the “ceiling” or carrying capacity is low, eventual extinction is possible by this mechanism. It is just a matter of waiting! The typical probability of extinction, per unit time (provided rK>>1, remember r=B-D) is given by: PE ≅ 2r 2 K ⋅ Exp[−rK ] , rK>>1 …(3-4) It says that there is no totally safe size of population, eventual extinction is certain for every size of population, since every value of K and r leads to a finite PE. As Keynes once said “We are all dead in the end”. Of course, the time may be very long, so it is a bit philosophical and this demographic mechanism alone seems insufficient to explain why even large populations suffer eventual extinction over periods of millions of years. The reason our calculations yield unreasonable results is that we have missed out the effects of environmental stochasticity. Thus extinction is a complex event involving the interaction of deterministic and random forces. • Even though the conditions imposed may be well-defined, the actual time of extinction is uncertain. • An organism may become extinct even in a “friendly” environment due to demographic stochasticity if the population is very low. Page - 24