Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 10, No. 2, 2024 

 

196 

Study	on	the	Gender	Change	of	the	Lamprey	in	
Ecosystem	Based	on	Verhulst	and	Logistic	Models	
Yingjie Wei1, †, *, Yuchen Fang1, †, Jing Cheng1, Zeduo Li1, Zongqian Yang1 

1College of Ocean and civil Engineering, Dalian Ocean University, Dalian, China 
† These authors also contributed equally to this work 
*Corresponding author: 15637461832@163.com 

 

Abstract: This paper focuses on the environmental impacts of changes in the sex ratio of the lampreys, the assessment of the 
strengths and weaknesses of lamprey’s adaptations, and their effects on ecosystem stability. In this paper, we first considered 
Verhulst and Logistic models of ecosystem-sex-ratio interactions, and predicted and analyzed the potential cascading effects of 
changes in lampreys sex ratio on larger ecosystems under different scenarios. Then, by integrating the Logistic model, we 
simulated the development and change of the ecosystem under various conditions, identified assessment metrics, and evaluated 
the relative strengths and weaknesses of the lampreys. The research in this paper provides important evidence for a deeper 
understanding of the role of the lampreys in ecosystems and the effects of changes in their sex ratio. 

Keywords: Verhulst Model, Logistic Model, Dynamic Model. 
 

1. Introduction 
In natural ecosystems, the sex ratio of males to females in 

most species’ populations will remain equal at 1:1, but the 
lampreys are different in that the proportion of sex change 
may deviate from this even ratio with changes in the external 
environment [1]. Scott et al [2] showed that the growth rate of 
the lampreys is affected by food, density, genetics, and the 
environment, and that the lampreys eel's growth rate 
determines its sex, so that ecological conditions determine the 
change of sex ratio of seven-gill populations. In this paper, the 
correlation between the sex ratio of lampreys and their 
specific living environment conditions was explored through 
mathematical modeling. 

First, this paper focuses on establishing Verhulst and 
logistic models for the sex ratio changes of lamprey’s 
populations under different natural marine environmental 
conditions, predicting and analyzing the potential cascading 
effects of the changes in the sex ratio of lampreys on larger 
ecosystems under different scenarios, and comparing the 
changes in the sex ratio and the environmental factors in the 
iterative process. The relative strengths and weaknesses of the 
lampreys were then assessed by modeling and comparatively 
analyzing food webs in lamprey’s habitat. By integrating the 
logistic model, the development and changes of the 
ecosystem under various conditions were simulated and 
assessment indicators were identified. The study in this paper 
provides new ideas for exploring and analyzing the 
correlation between population growth and changes in 
population sex ratio of the lampreys and specific conditions 
in the ecosystem. 

2. Racial Dynamic Model 

2.1. Population dynamic models 
The dynamic model for the population size of lampreys is 

established to describe the growth of population size under 
limited resource conditions. 𝑁 is defined as the population 
size of lampreys and the Verhulst model is established [3]. 
The formula is as follows: 

𝑟 𝑃𝑓 𝑛 1 𝛼            (1) 

 

𝑟 𝑃 𝑛 1 𝛼            (2) 

 
In the above equation, 𝑟   is the inherent growth rate of 

female lampreys. 𝑟   is the inherent growth rate of male 
lampreys. 𝑃𝑓 𝑛   is the proportion of lamprey’s females, 
𝑃 𝑛   is the proportion of lamprey’s males, and 𝐾  is the 
maximum average biomass of species within the range of 
environmental restrictions. 𝛼𝐹  is the saturation parameter 
for the growth of female lampreys, on the contrary, 𝛼   is 
males. 𝑛 is the number of iterations. 

2.2. Logistic model for lamprey sex ratio 
The logistic model for lamprey’s sex ratio is established [4], 

which could analyze the impact of food supply on the 
developmental rate of juvenile lampreys and changes in the 
sex ratio of the lamprey’s population. The formula is as 
follows: 

 
𝑃 𝑛                (3) 

 
In the above equation, 𝑃 𝑛  represents the possibility of 

juvenile lampreys developing into females. Parameter 𝑟  is 
the development rate, and 𝑛  is the time when the sex ratio 
of lampreys begins to change. 

Considering the complexity of the sex determination 
mechanism in lampreys, a more detailed model needs to be 
constructed. The environmental factor 𝑇  is introduced as a 
key regulatory parameter to expand the model, which can 
more comprehensively reveal how environmental conditions 
affect changes in the sex ratio of lamprey’s populations. 

 
𝑃 𝑛              (4) 

 
In the above equation, 𝑟 𝑇  represents the effect of 𝑇 on 

the development rate of lampreys. 



 

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2.3. Ecosystem model 
Through in-depth research and application of models for 

the population size and sex ratio of lampreys, it is possible to 
clarify the changes in population size and sex ratio of 
lampreys under different specific living environment 
conditions, as well as how these changes further affect a wider 
ecosystem. 

2.3.1. Changes in the ecological balance 
In order to explore the impact of changes in the sex ratio of 

the lamprey population on the quantity and distribution of 
other biological groups, the food supply and male sex ratio of 
the lamprey population were included in the model [5]. The 

ratio of initial food resources to the male population of 
lampreys is defined as the food supply. The formula is as 
follows: 

 

𝐹                  (5) 

 
In the above equation, 𝐹  is the food supply and 

influencing factor, and 𝑀  is the initial amount of food 
resources. The initial food resource is 100, and the change in 
male to female sex ratio from 0 to 1 was simulated. The results 
are shown in Figure 1. 

 

 
Figure 1. The effect of food supply on lamprey’s sex ratio 

 
Figure 1 shows that as the food supply decreases, the 

proportion of females in the lamprey population significantly 
decreases. The food supply has a significant impact on the sex 
ratio of the lamprey population. When the food supply 
decreases to a certain extent, the population of lampreys is 
entirely male. In this extreme situation, due to the lack of 
female individuals, the population is unable to engage in 
normal reproductive activities, resulting in a sharp decrease 
in population size. 

2.3.2. The impact of the food chain 
In order to explore the impact of gender ratio changes in 

the population of lampreys on the number of predators, the 
following equation is established: 

In the case of sufficient food: 
 

𝑃 𝑛 𝑃 𝑛 1 𝛽           (6) 

 
𝑃𝑡 𝑛 𝑃𝑡 𝑛 1 ∙ 1 𝜃           (7) 

 
In the case of food shortage: 
 

𝑃 𝑛 𝑃 𝑛 1 𝛾            (8) 
 

𝑃𝑡 𝑛 𝑃𝑡 𝑛 1 ∙ 1 𝜉            (9) 
 
In the above formula, 𝑃𝑡 𝑛  is the number of predators. 

𝛽 is the rate of increase in the male proportion of lampreys. 
𝛾  is the rate at which the male proportion of lampreys 
decreases. 𝜉 is the rate at which predators increase, and 𝜃 is 
the rate of predator reduction. Assuming 50 iterations, the 
resulting line graph is in Figure 2. 

 

  
Figure 2. Changes in sex ratio of lampreys and their impact on predator numbers 

 
Figure 2 shows that when food resources are abundant, over time, the male sex ratio of lampreys decreases, and the 



 

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number of predators increases. When food resources are 
scarce, the male sex ratio of lampreys increases, and the 
number of predators decreases. Therefore, under sufficient 
food conditions, there is a positive correlation between the 
proportion of females in the lamprey population and the 
number of predators. 

2.3.3. The impact of reproductive success 
Due to the influence of various factors on the sex ratio of 

the lamprey population during each iteration, the formula is 
adjusted to: 

 
𝑃 𝑛 1 𝑚𝑖𝑛 1, 𝑚𝑎𝑥 𝑃 1 , 𝑃 𝑛 𝜑     (10) 

 
In the above equation, 𝜑 represents a random number with 

a standard normal distribution and establish a model: 

 
𝑅𝑎 𝑛 1 𝑚𝑖𝑛 1, 𝑚𝑎𝑥 𝑅𝑎 1 , 𝑅𝑎 𝑛 𝜑    (11) 

 
In the above equation, 𝑅𝑎 𝑛  is the resource availability 

rate, and the success rate of reproduction (𝑅𝑠) of lampreys is 
calculated based on the sex ratio of the population. Assuming 
that the success rate of reproduction of lampreys is directly 
proportional to the change in population sex ratio, the formula 
is expressed as: 

 
𝑅𝑠 1 𝑎𝑏𝑠 𝑃 𝑛 𝜗            (12) 

 
In the above equation, 𝜗 Indicates the proportion of male 

sex of lampreys at the optimal reproductive rate. The final 
simulation diagram is shown in Figure 3. 

 

 
Figure 3. The impact of the change of lamprey’s sex ratio on their own reproduction 

 
As shown in Figure 3, the success rate of reproduction of 

lampreys is closely related to changes in their population sex 
ratio. As available resources increase, the proportion of 
females in the lamprey population increases and the success 
rate of reproduction decreases. Therefore, resource changes 
can lead to changes in the sex ratio of the lamprey population, 
thereby affecting the success rate of lamprey reproduction. 

2.3.4. Changes in habitat use 
Firstly, the sex ratio of the lamprey population is simulated, 

then the habitat occupation of the lamprey is simulated in each 
generation. Finally, simulation analysis is conducted by 
controlling the growth rate of its sex ratio [6]. The formula is 
as follows: 

 
𝑃 𝑛 1 𝑃𝑓 𝑛              (13) 

 
𝑃𝑓 𝑛 𝑃𝑓 𝑛 1 𝜑            (14) 

 
𝑂𝑝 𝐻𝑐 ∙ 𝑃𝑓 𝑛 𝜀             (15) 

 
In the above equation, 𝑃𝑓 𝑛   is the proportion of 

lamprey’s females after n iterations. 𝑂𝑝 is the proportion of 
habitat occupation, and 𝐻𝑐 is the total capacity of the habitat. 
𝜀 is other influencing factor. The results are shown in Figure 
4. 

 

  
Figure 4. The impact of the change of lamprey’s sex ratio on their habitat use 



 

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Without limiting resources, Figure 4 shows that the 
proportion of male sex in the lamprey population increases 
and the habitat area decreases. When the habitat area reaches 
its lowest point, the male sex ratio of the lamprey species 
reaches its maximum habitat capacity. 

Assuming limited resources, interspecific competition is 
also one of the factors affecting habitat area. this situation is 
simulated. The formula is as follows: 

 
𝑅𝑟 𝑛 𝑅𝑟 𝑛 1 𝜑            (16) 

 
𝑂𝑝 𝐻𝑐 ∙ 𝑃𝑓 ∙ 𝑅𝑟 𝑛               (17) 

 
In the above equation, 𝑅𝑟 𝑛   is the proportion of 

resources. The results showed that the proportion of male 
population in lampreys increased, while the habitat area and 
resource share decreased. From the results of 100 iterations, 
it can be seen that the proportion of male population and 
resource share of lampreys are generally decreasing. The 
higher the proportion of male population, the lower the habitat 
area. 

2.3.5. Ecosystem stability 
By simulating the change of seven species of lamprey’s sex 

ratio in ecosystem, the comprehensive index is calculated. the 
following Figure 5 is obtained. 

 

 
Figure 5. The change of the change of lamprey’s sex ratio 

in ecosystems 
 
Figure 5 shows that when the sex ratio of the lamprey 

population changes, it will have a chain reaction on the 
balance of the entire ecosystem. 

 

 
Figure 6. The impact of the change of lamprey’s sex ratio on ecosystem stability 

 
Figure 6 shows that the sex ratio of the lamprey population 

changes, and its population size and reproductive success rate 
also change. The predator number of lampreys and their own 
feeding habits will adjust accordingly, which will affect other 
species in the local ecosystem. In some cases, imbalanced sex 
ratios in lamprey populations may lead to a decrease in the 
stability of local ecosystem resistance. 

3. Food Web Model 
The development and gender ratio changes of lamprey 

populations are simulate under different ecosystem conditions. 
The formula is as follows: 

 

𝑃 𝑛 𝑃 𝑛 1 𝐿𝑓             (18) 
 

𝐿𝑓 𝑚𝑎𝑥 0.1𝑃 𝑛 , 𝑚𝑖𝑛 0.1𝑃 𝑛 , 𝜑       (18) 
 

𝑃 𝑛 𝑚𝑎𝑥 0, 𝑚𝑖𝑛 𝑃 𝑛            (19) 
 
The above equation indicates that the change in the male 

sex ratio of each generation of lampreys depends on the male 
sex ratio of the previous generation. 

To limit fluctuations, the male sex ratio of the lamprey 
population can be controlled within 10% of the initial value. 
The following formula can be derived: 

 
𝑅𝑠 𝑛 𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑒 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑟𝑎𝑡𝑒 ∙ |1 𝑃 𝑛 | (20) 



 

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The success rate of reproduction is influenced by the male 
sex ratio of the lamprey population. 

 
𝑃𝑡 𝑛 𝜏 ∙ 𝑃𝑡 𝑛 1 ∙ 𝑅𝑠 𝑛 1         (21) 

 
In the above equation, 𝜏 is the survival rate of lampreys. 
 

𝐹𝑟 𝑛 𝑚𝑖𝑛 1, 𝑚𝑎𝑥 0, 𝑅𝑒𝑠𝑜𝑢𝑟𝑐𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑡 𝑛 (22) 
 

 𝐶𝑖 𝑤_𝑟𝑎 ∙ 𝑃 𝑛 𝑤_𝑟𝑒 ∙ 𝑅𝑠 𝑛 𝑤_𝑝𝑟 ∙ 𝑃𝑡 𝑛
𝑤_𝑓𝑑 ∙ 𝐹𝑟 𝑛                (23) 

 

In the above equation, 𝐹𝑟 𝑛  is the limited food resource. 
"𝑤_𝑟𝑎", "𝑤_𝑝𝑟", "𝑤_𝑟𝑒", and "𝑤_𝑓𝑑" represent the weights 
of various variables such as the male sex ratio, reproductive 
success rate, predator number, food, and resources of the 
lamprey population, respectively. This formula represents the 
calculation of comprehensive indicators, which can be used 
to evaluate the state of the entire ecosystem. The weight can 
be adjusted according to the actual situation to reflect the 
relative importance of different factors. 

By simulating the male dominant ratio, female dominant 
ratio, and male to female equal ratio in the population of 
lampreys, a radar chart is obtained, as shown in Figure 7. 

 

  
Figure 7. Radar chart of lamprey’s sex ratio under different ecosystem conditions 

 
Figures 7 indicates that ecosystem provider is the main 

advantage of lampreys. At the same time, the advantage of 
female dominance is greater than that of male dominance and 
gender equality, while the latter two are basically the same in 
their evaluation advantages. It shows that they provide a 
variety of ecological services and contributions to the entire 
ecosystem. These services and contributions are of great 
significance for maintaining ecological balance, stability and 
biodiversity. On the whole, in the three cases, the dominant 
disadvantage is environmental pressure, and the following 
disadvantages need to be discussed in three cases. The first is 
the average gender ratio, followed by reproduction threat. 
When the male is dominant, Predation Threat is a secondary 
disadvantage, and finally the female is dominant, Overfishing 
Risk is a secondary disadvantage. 

4. Conclusion 
The aim of this paper is to investigate the environmental 

impacts of changes in the sex ratio of the lampreys, the 
assessment of the strengths and weaknesses of sex adaptations 
in the lampreys, and their impacts on ecosystem stability. First, 
Verhulst and logistic models of sex ratios in lampreys’ 
populations were developed to predict and analyze the effects 
of variation in lampreys sex ratios on larger ecosystems under 
different scenarios. It was found that while variability in the 
sex ratio of lampreys enhanced their adaptability, it also 
increased the uncertainty of the ecosystem as a whole. Second, 
the relative strengths and weaknesses of lampreys were 
assessed by modeling and comparatively analyzing food webs 
in their habitats. It was found that the main advantage of the 

lampreys was its role as an ecosystem provider, while the 
main disadvantage was environmental stress, and secondary 
advantages and disadvantages also varied by sex ratio. This 
paper investigates the importance of the balanced sex ratio of 
the lamprey’s population for the maintenance of ecological 
balance, stability and biodiversity. 

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