Extinction probabilities as a function of temperature for populations of tsetse (Glossina spp.)

Abstract

Significant reductions in populations of tsetse (Glossina spp) in parts of Zimbabwe have been attributed to increases in temperature over recent decades. Sustained increases in temperature might lead to local extinctions of tsetse populations. Extinction probabilities for tsetse populations have not so far been estimated as a function of temperature. We develop a time-homogeneous branching process model for situations where tsetse live at different levels of fixed temperature. We derive a probability distribution pk(T) for the number of female offspring an adult female tsetse is expected to produce in her lifetime, as a function of the fixed temperature at which she is living. We show that pk(T) can be expressed as a geometric series: its generating function is therefore a fractional linear type. We obtain expressions for the extinction probability, reproduction number, time to extinction and growth rates. The results are valid for all tsetse, but detailed effects of temperature will vary between species. No G. m. morsitans population can escape extinction if subjected, for extended periods, to temperatures outside the range 16°C-32°C. Extinction probability increases more rapidly as temperatures approach and exceed the upper and lower limits. If the number of females is large enough, the population can still survive even at high temperatures (28°C-31°C). Small decreases or increases in constant temperature in the neighbourhoods of 16°C and 31°C, respectively, can drive tsetse populations to extinction. Further study is needed to estimate extinction probabilities for tsetse populations in field situations where temperatures vary continuously.

Fig 1. Daily mortality rates for female G. m. morsitans pupae for temperatures in the range (15°C–33°C).

Eq (9) plotted for different values of temperature.

Fig 2. Daily mortality rates for young adult and mature adult female G. m. morsitans for temperatures ranging from 15°C–35°C.

Eqs (10) and (11) plotted for different temperatures.

Fig 3. Probability of extinction of a population of G. m. morsitans as a function of temperature varying between 15°C and 35°C.

Eq (7) solved for different values of temperature and for different numbers of inseminated adult females in the initial population.

Fig 4. Expected number of surviving female offspring per adult female G. m. morsitans for different temperatures (15°C–35°C).

Eq (5) solved for different values of temperature. The fine vertical dotted line at about 19°C indicates the temperature at which a female produces the greatest number of surviving daughters. The dashed vertical lines just below 15°C and just above 31°C, respectively, indicate the temperatures below and above which a female produces fewer than one surviving daughter.

Fig 5. Expected number of generation to extinction of a G. m. morsitans population for different number of females in the initial population at different temperatures (15°C–35°C).

Eq (8) is solved iteratively up to n = 20.

Fig 6. Expected growth in the numbers of adult females in a G. m. morsitans population at different temperatures (15°C–35°C).

The projections are approximately valid for the early stages of growth, before density dependent processes have noticeable effects.

Fig 7. Daily growth rate (%) of a population of G. m. morsitans living at different constant temperatures (15°C–35°C).

The vertical dotted line at 25°C indicates the temperature at which the growth rate is maximised.