Yiheng LIU

Xi’an Jiaotong University,Xi’an 710049,China

The Prediction of Population of the Yangtze Finless Porpoise

Yiheng LIU*

Xi’an Jiaotong University,Xi’an 710049,China

[Objective] The aim was to study the current living situation of Yangtze finless porpoise,and explore the prediction of the number of Yangtze finless porpoise in next few years.[Method]Modified Leslie Model and Siler’s Model was applied to estimate the number of Yangtze finless porpoise.[Result]Using the model, Yangtze finless porpoise will extinct in nearly 20 years.[Conclusion]The study provides a cautionary warning of the conservation of the freshwater cetacean species.

Leslie Model;Siler’s Model;Mortality rate;Quantify human activity

T he research focused on the Yangtze finless porpoise,a highly endangered animal species living in the Yangtze River and it has driven a heated discussion on biological and environmental in China in recent years.

In the year 2006,almost 40 scientists from all over the world gathered together on the Yangtze River for the purpose of searching the Yangtze River dolphin but they did not find the trail of them.In 2007,the Yangtze finless porpoises become the only cetacean species living in the Yangtze River. The Yangtze finless porpoise is a subspecies of the narrow-ridged finless porpoise with its habitats distributing over the middle and lower reaches of the Yangtze River.The lovely appearance wins her a name-fairy of the Yangtze River.However,progressive declines in the number of porpoises observed during boat surveys,drastic loss of suitable habitats resulting in apparent distribution gaps in the formerly contiguous population,and low levels of genetic diversity all suggest that the Yangtze finless porpoise may follow the Yangtze River dolphin(Baiji) and also become extinct in the wild in the near future.

As known,the only home of the Yangtze finless porpoise-the Yangtze River is known as the“Golden Channel”of central China.It has been supporting the livelihoods of hundreds of millions of people through agriculture, aqua-culture and industrial activities. Unfortunately,the noise generated by high-frequency shipping transportation,the application of illegal fishing equipment,and the existence of many water power facilities such as the Three Gorges Dam,all posed great threats on the lives of Yangtze finless porpoise.There an apparent decline of porpoise population in the Yangtze mainstem,from more than 2550 animals in 1991 to fewer than 1225 animals in 2006,raising an important question:how many years do we have left to reserve the decline of this lovely cetacean?

So it is urgent for us to learn more about the survival status of the Yangtze finless porpoises by estimating the number of this species now and predict the tendency of the population of the Yangtze finless porpoises of the coming years.In order to solve this question,we have an incentive to know the actual rate of porpoise decline.So,to get the most reliable results,we should use the scientific data and set up a mathematical model to predict the change of the population.

Methods

The Leslie model and preliminary analysis

In some species,the amount of reproduction varies greatly with the age of individuals.For instance,consider two different human populations that have the same total size,if one is comprised primarily of those over 50 in age,while the other has mostly individuals in their 20 s,we would expect quite different population growth from them.Clearly,the age structure of the population matters.Humans progress through a relatively long period before puberty when no reproduction occurs. After puberty,various social factors discourage or encourage childbearing at certain ages.Finally,menopause limits reproduction by older women.

To capture the effects on population growth,it is necessary for modeling a human population by creating five age classes with:

x1(t)=No.of individuals age 0 through 14 at time t,

x2(t)=No.of individuals age 15 through 29 at time t,

x3(t)=No.of individuals age 30 through 44 at time t,

x4(t)=No.of individuals age 45 through 59 at time t,

x5(t)=No.of individuals age 60 through 75 at time t.

Although this formulation makes the unrealistic assumption that no one survives past age 75,that shortcoming could of course be remedied by creating additional age classes.Using a time step of 15 years,we can describe the population through equations like:

Here,fidenotes a birth rate(over a 15-year period)for parents in the ithage class,and τi,i+1denotes a survival rate for those in the ithage class passing into the(i+1)th.Because a single set of parents may be in different age groups,we should attribute half of their offspring to each in choosing values for fi.

In matrix notation,the model is simply xt+1=Pxt,where

is the column vector of subpopulation sizes at time t and

is the projection matrix.

In the Yangtze finless porpoise species,the amount of reproduction varies greatly with the age of individuals,so the Leslie model is suitable for the estimation of the Yangtze finless porpoise species population.

However,there is a critical problem should be solved before the next step-the data are not explicit enough. All age-at-death data(n=279 animals) used in the analysis came from stranded porpoises that were opportunistically collected along the middle and lower reaches of the Yangtze River.And only their body length and gender are known.By looking up the historical papers of body features of the finless of the finless porpoise samples,Zhang[1]used the biological information(including their gender and body length)of 68 samples to derive an age-length curve.Generally,the relationship between age X and body length(in cm)of male(denoted Lm) and female(denoted Lf)Yangtze finless porpoises can be expressed by the following two equations:

Thus an explicit age distribution of samples were obtained.

Mortality rates:Siler’s Model

Another important concern for Leslie Matrix model is to figure out the survivorship(or mortality)rates of Yangtze finless porpoise.Due to the huge differences among the history records of survivorship,these calculated historical parameter values would not be applied directly.On the contrary,Siler’s Competing-Risk Model of Survivorship[5]was applied to estimate the survivorship of the Yangtze finless porpoises.Siler’s model was used because it can accurately describe mammalian survivorship and mortality schedules[5-7].

In Siler’s Model,age-specific survivorship l(x)is determined by three components:

Including an exponentially decreasing risk of mortality due to juvenile risk factors,

a constant risk experienced by all age classes,

and an exponentially increasing risk due to senescent risk factors:

The five Siler’s parameters(hazard weight,a1,a2and a3;adjustment constant,b1and b2)were regressed by the least-squares method.

To get the 5 parameters a1,a2,a3, b1,b3,the Most-Likelihood-Estimates method should be used.And the L(n) is the likelihood function.

In the Leslie model,it can be expressed as:

Demographic rates

The birth rates derived from samples is another important factor of our Leslie model.But in consideration of the complexity of the existing birth rate models and also the limitation of our tools and knowledge,it is too difficult to directly calculate the birth rate of the Yangtze finless porpoise.After looking up a lot of related materials online,it found that according to existing biological information,finless porpoises breed every October and they have only one offspring in each embryo.So, it is assumed that every female finless porpoise has one offspring per year. No doubt it is known that it is not very realistic and there are many other factors that have influence on the birth rates of the finless porpoise(e.g.environmental change,climate,human activities,etc.).Assuming that the ratio of new-born porpoises to female porpoises must be lower than 1,and the original purpose for us is to estimate how long the Yangtze finless porpoise will become extinct,even if we assume the rate is 1 which may be much larger than the real ratio,the high possibility of the extinction of the Yangtze finless porpoise can be concluded.And thefollow is our material.

Table 1The material used to estimate Yangtze finless porpoise

From the table before,it can be concluded that the mutuality age of Yangtze finless porpoise is 4.5 for male and 4 for female.And according to the research of Prof.Zhang,it is known that Yangtze finless porpoise obey monogamy and they have only one offspring in a year.So we imagine that every couple from 4-year-old to 16-year-old have on offspring in a year.We can write as:

Table 2Comparison among various study results on the age and body length of maturity inN.P.αsiaeorientalis

xi(t)can be defined as mentioned before.

And

xm,i=No.of the male Yangtze finless porpoise of agei,

xf,i=No.of the female Yangtze finless porpoise of agei,

So in the Leslie model can be expressed as:

Gap:quantification of Human Impact

After the preliminary work has been completed,the results were shown below:

After calculating,it can be found there are some“gaps”between the numbers from calculating and a series of real data,we want to find the reason.

As known,the anthropogenic impacts on the population of Yangtze finless porpoise cannot be ignored as the economic development has been doing great harm to the habitats of animal.So it would explain the existence of the gap by the impact of increasingly worsening human activity impact, especially from economic development.Since the economic reform policies have been accompanied by largescale environmental development programs such as dam construction and land reclamation.

The research uses the gap between the calculation results for year 2002 and the real data in 2002(the middle year between 1993 and 2014) as a constant quantified degree of human impact.What is worth mentioning here is the gap is dealt with by simply uniformly apply it on every age class. That is,the gap for each age class equals the total gap divided by the number of age class.We can express as follows:

For the constant quantified degree:

x2002=the real data collected in 2002,x2002=the data calculated in 2002

Table 3Life table parameters,including number(nx)and survivorship(lx)at agexbased on 213 sample

For demographic rate

After improving,the Leslie model can be expressed as follows:

Results

Surprisingly,after considering the human activity impact,the estimated extinction time of the Yangtze finless porpoise will be approximately 20 years later——the year 2025.However,uncontrolled human impact will only be worse as time passes while birth ratio equals 1.So the accurate extinction date of Yangtze finless porpoise might be even earlier!

Discussions

Uncertainties about the age distribution of samples

All age-at-death date(n=279 animals)used in the analysis came from stranded porpoises which were opportunistically collected along the middle and lower reaches of Yangtze River.It is very difficult to get an accurate age distribution.The historical papers of body features of the finless porpoise samples are looked up.And Zhang[1]used the biological information of 68 samples(including their sex and body length)to derive an age-length curve. Generally,the relationship between age x and body length(in cm)of male(denoted Lm)and female(denoted Lf)Yangtze finless porpoises he claimed can be expressed by two simple equations mentioned above in the methods part.It is,therefore,estimated the age of these additional 213 animals using the equations shown above.

Uncertainties about the survivorship(or mortality)rates of finless porpoise

As for Leslie Matrix model,it is important for us to know is the survivorship(or mortality)rates of Yangtze finless porpoise.Due to the huge differences among the history records of survivorship,these calculated parameter values are not directly applied.Instead,Siler’s competing-risk model of survivorship[5]was used to estimate the survivorship of finless porpoises,because it accurately describes mammalian survivorship and mortality schedules[5-7].

The main challenge——birth rates

The birth rates derived from samples is another important factor of the Leslie model.But in consideration of the complexity of the existing birth rate models and also because of the limitation of our tools and knowledge,it is too difficult for us to directly calculate the birth rates of the Yangtze finless porpoise.According to existing biological information,finless porpoises breed every October and they have only one offspring in each embryo.So, we simply assume that every female finless porpoise has one offspring per year.We know it is not very realistic and there must exist many other factors influencing the birth rates of finless porpoise.That is,the ratio of newborn porpoises to female porpoise must be lower than 1.However,the original purpose for us is to estimate how long the Yangtze finless porpoise will become extinct.So,even if we assume the rate is 1,which may be much larger than the real one,we can still how fast the finless porpoises will disappear.

Progress——with consideration of human activity

As we all know,the anthropogenic impact on the population of finless porpoises cannot be ignored as the economic development sometimes does great harm to the habitats of animals.When looking up other people’s work on the population of the finless porpoise,it is found few of them considered human activities as an important factor.To estimate this impact, the research used the data calculated from our model to compare with the historical data of population.A gap (value of difference per 4 years)between them has been found.As shown before,the data from 1978 to 2012 were collected.However,since the economic reform policy launched in 1992 by Deng Xiaoping,economic and industrial activities along the Yangtze River have grown exponentially,and have been accompanied by largescale environmental development programs such as dam construction and land reclamation.What’s more,systematic range-wide abundance estimates for the Yangtze finless propose were also first published in 1993.So, the data before 1993 were omitted and only data from 1993 to 2012 were used.The research uses the gap between the calculation results for year 2002 and the real data in 2002(the middle year between 1993 and 2014) as a constant quantified degree of human impact.

The drastic decline of the Yangtze finless porpoise population as a result of rapid industrial and economic growth reveals how uncontrolled human impact threatens the future of freshwater cetaceans.Our study provides a cautionary warning of the con-servation of the freshwater cetacean species.

Problems in Matlab

When explored 5 parameters for Siler’s model with Most-Likelihood-Estimates,it found that the final result will change with the initial vector changing.The research studied it for a while,but it can hardly find the reason. However,the research have several estimates.The most believable one is the function is too complex for matlab to find an explicit solution.For examine this suppose,gatbx tools are used to solve the problem.The results suggest that the results are the local minimums.

To some certain extent,it’s proved our conjecture,but it cannot pretty sure about the relationship between the two results.

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Responsible editor:Xiaoxue WANG

Responsible proofreader:Xiaoyan WU

*Corresponding author.E-mail:liuyiheng.cool@163.com

Received:May 4,2015 Accepted:June 16,2015