A method and system for analyzing population numbers in a food chain based on an iterative system
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHANDONG UNIV OF SCI & TECH
- Filing Date
- 2026-03-09
- Publication Date
- 2026-06-09
Smart Images
Figure CN122177199A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of ecological technology, and in particular relates to a method and system for analyzing population size in a food chain based on an iterative system. Background Technology
[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.
[0003] The analysis of population growth rate changes in food chains has broad and significant applications in ecology. In environmental protection, scientific population size assessment is fundamental to the formulation and implementation of species conservation strategies. For example, by calculating population sizes in food chains, the health of ecosystems can be assessed, guiding the planning and management of nature reserves. Simultaneously, in ecological restoration projects, population sizes can be used to set restoration targets, assess restoration effectiveness, and optimize restoration pathways, thereby improving the targeting and effectiveness of ecological restoration. In species management, population size analysis in food chains directly relates to the practical effectiveness of endangered species protection and invasive species control. For endangered species, population size and structure analysis can identify their main threats, assess extinction risks, and design scientific breeding and release programs, habitat restoration, and other protection measures. For invasive species, population dynamics models can be used to predict their spread trends, assess ecological and economic impacts, and formulate timely and effective control plans to reduce their negative impacts on local biodiversity and ecological security.
[0004] Existing population calculation methods typically employ a single second-order nonlinear differential equation, which can only analyze the rate of change of a single population's population growth rate. They cannot accurately describe the influence of population growth rate changes across the entire food chain on the population changes of other species within the same food chain. Furthermore, existing methods neglect the impact of changes in the population's own size on the rate of change of growth rate, leading to inaccurate population calculations. Summary of the Invention
[0005] To overcome the shortcomings of the existing technologies, this invention proposes an analysis method and system for population size in food chains based on an iterative system. This method is applicable to the analysis and calculation of the population growth rate of food chains in environmental protection and species management. By using an iterative system, the population size calculation process takes into account the influence of the population growth rate of the entire food chain and the population size of the species itself, thus providing a more comprehensive and accurate description of population size changes. It can capture the influence of population changes of other species and the population size of the species itself, which are ignored in traditional methods, thereby improving the accuracy and applicability of the model.
[0006] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions: In a first aspect, this invention discloses a method for analyzing population size in a food chain based on an iterative system, comprising: Obtain population information within the target food chain; A food chain population iterative system is constructed based on a second-order iterative system with non-homogeneous boundary conditions. The iterative system incorporates the predator-prey relationship between species into the equation and has non-homogeneous boundary conditions. By using the Guo-Krasnosel' fixed-point theorem and the range of eigenvalues in the Banach space, it is proved that the iterative system has at least one positive solution; Based on the positive solution, the population changes of each species in the food chain are analyzed.
[0007] Secondly, this invention discloses an analysis system for population size in a food chain based on an iterative system, comprising: The data acquisition module is configured to acquire population information in the target food chain. The model building module is configured to: construct a food chain population iterative system for the food chain based on a second-order iterative system with non-homogeneous boundary conditions, wherein the iterative system incorporates predation and predation relationships between species into the equation, and the iterative system has non-homogeneous boundary conditions. The positive solution proof module is configured to: use the Guo-Krasnosel' fixed-point theorem in combination with the eigenvalue range on the Banach space to prove that the iterative system has at least one positive solution; The change analysis module is configured to analyze the population changes of each species in the food chain based on the positive solution.
[0008] Thirdly, the present invention discloses an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, complete the steps of the above-described method for analyzing population size in a food chain based on an iterative system.
[0009] Fourthly, the present invention discloses a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the above-described method for analyzing population size in a food chain based on an iterative system.
[0010] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention uses a second-order iterative system to simultaneously consider the changes in population size, the rate of change, and the rate of change of the rate of change, and can capture the direct and indirect influences between adjacent species in the food chain, making the model closer to ecological reality.
[0011] This invention systematically applies the Guo-Krasnosel'skii fixed-point theorem to a food chain population dynamics model, rigorously proving the existence of the positive solution under broad conditions, and solving the problem of weak mathematical foundation in traditional ecological models.
[0012] The non-homogeneous boundary conditions and environmental factors or population intrinsic growth coefficients introduced in this invention can more flexibly simulate specific initial / final state conditions and the intensity of interactions between species, thereby improving the accuracy of the simulation.
[0013] The analytical method proposed in this invention can be directly applied to specific ecological conservation practices, such as predicting the recovery trajectory of endangered species populations, assessing the spread potential of invasive species, and optimizing species configuration schemes in ecological restoration projects.
[0014] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description
[0015] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.
[0016] Figure 1 This is a flowchart of the method for analyzing population size in a food chain based on an iterative system, as described in Embodiment 1 of the present invention. Detailed Implementation
[0017] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.
[0018] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.
[0019] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.
[0020] Example 1 In one or more embodiments, a method for analyzing population size in a food chain based on an iterative system is disclosed, such as... Figure 1 As shown, it includes the following steps: Step S1: Obtain population information in the target food chain. Population information includes, but is not limited to: (1) Time series observation data on the population size of each species, such as historical monitoring data and survey statistics; (2) Environmental factor data, including but not limited to external conditions that may affect population growth, such as temperature, precipitation, habitat area, and pollutant concentration; (3) Predator-prey relationships between species, such as food web structure and trophic level connections; (4) Population size of each species at the initial moment ,Right now ; (5) Population size set at the target time (e.g., the end of the management period). ,Right now , used to simulate ecological restoration or protection goals; (6) Species’ intrinsic growth potential parameters, including but not limited to basic biological parameters such as reproductive rate and mortality rate.
[0021] Step S2: Construct a food chain population iterative system for the food chain based on a second-order iterative system with non-homogeneous boundary conditions. The iterative system incorporates the predator-prey relationship between species into the equation, and the iterative system has non-homogeneous boundary conditions.
[0022] The expression for a second-order iterative system with non-homogeneous boundary conditions is: (1) In the formula, Let be the population size of the i-th species in the food chain at time t; It is the second-order rate of change of population size, reflecting the acceleration or deceleration trend of population growth. It is often used to describe the combined effects of factors such as environmental pressure, food supply, and the influence of natural enemies on a population. It is an environmental factor or the intrinsic growth coefficient of a population, reflecting the growth potential of the species in the food chain, and is affected by factors such as climate and habitat quality; Let be the interspecies interaction function, representing the influence of the growth rate of the i-th species on the growth rate of its next trophic level. The impact; To represent a closed food chain, meaning there is a feedback relationship (such as energy cycling or matter cycling) between the apex predator and basic producers, m is the total number of species in the food chain, i.e., the number of trophic levels from producers to the apex predator. In a closed food chain, And satisfy , of which There is a feedback relationship between the first species (the apex predator) and the first species (the producer).
[0023] Traditional models often analyze single species independently, neglecting the interactions between upstream and downstream species in the food chain. This example uses... By incorporating predator-prey relationships between species into the equation, the iterative system captures the dynamic connections between species, thus more realistically simulating the complexity of ecosystems.
[0024] Furthermore, the second-order iterative system has non-homogeneous boundary conditions as follows: (2) (3) In the formula, The initial population size of species i at the time starting point (e.g., the time at which observation begins); This represents the target population size of species i at the end of the time period (e.g., the end of the observation period), which can be used to set ecological restoration goals or simulate population status under different management strategies.
[0025] Using equations (1)-(3) above, this embodiment constructs a model to simulate the real interactions within a food chain. Equation (1) quantifies the predator-prey relationship between species. It shows that species... The acceleration of population change (which can be understood as the "driving force" or "pressure" of population growth) depends on the species at the next trophic level. The number of animals. For example, the rate of change in the wolf pack (predator) population is directly affected by the number of sheep (prey). This is more realistic than traditional models that only consider a single species. The boundary conditions in formulas (2)-(3) directly incorporate ecological management objectives into the model. This refers to the current initial number of species (survey data), while It is used to express the hope that at a certain point in the future (e.g.) The model can not only predict the target number (such as the target population in the protected area planning) but also be used for planning.
[0026] This embodiment supports ecological management objectives by setting boundary conditions. and The model can be used to simulate protected area planning, set target population sizes, evaluate the effectiveness of different conservation measures, predict population recovery paths, and simulate the recovery process of endangered species by adjusting boundary conditions.
[0027] Step S3: Using the Guo-Krasnosel' fixed-point theorem combined with the eigenvalue range on the Banach space, prove that the iterative system has at least one positive solution.
[0028] This embodiment proves the existence of the correct solution, wherein , and It is an integer. This embodiment, based on existing research on the existence of positive solutions to second-order iterative systems, proves the existence of the positive solution to the iterative system by giving the corresponding Green's function and its related properties, and using the Guo-Krasnosel'skii fixed-point theorem.
[0029] First, define the limit growth rate parameter as: (4) In the formula, When the population size is close to zero, it represents the endangered status of the species and reflects the species' maximum recovery potential when its numbers are extremely low. When the population size approaches infinity, it represents a state of species explosion, reflecting the minimum growth potential of a species when its population size is extremely large. For a time sub-interval, and It is used to eliminate transient fluctuations at the beginning and end of the observation period (such as disturbances at the beginning of the survey and anomalies at the end), and to focus on the dynamics of the stable phase of the ecosystem.
[0030] Formula (4) defines the limit states: quantifying the "resilience" and "threat" of a species. Endangered status measures a species' recovery potential when its numbers are extremely low (e.g., endangered species). The larger the value, the stronger the species' ability to recover by relying on its own reproduction and utilizing limited resources when facing extinction. Explosive status measures a species' continued expansion potential when its numbers are extremely high (e.g., invasive species). The larger the value, the more difficult it is for the species' growth momentum to be suppressed by natural enemies or resource shortages during an explosive state.
[0031] In this embodiment, the limiting growth rate parameter is defined to reveal the extreme state behavior of a population and the endangered status of a species. It can be used to assess the natural recovery capacity of endangered species; the higher the value, the stronger the species' recovery potential in an endangered state. In practical applications, it can be used to prioritize species conservation efforts. (Species outbreak status) It can be used to assess the continued expansion capacity of invasive or dominant species. The higher the value, the stronger the potential for continued growth of the species in an outbreak state. In practical applications, it can be used to provide early warning of the risk of ecological imbalance.
[0032] This embodiment makes the following assumptions: (C1)
[0033] (C2)
[0034] In the formula, The set of positive real numbers, i.e., all real numbers greater than zero, is typically used in population size models to represent non-negative population sizes. ; A continuous function space, specifically defined on a closed interval. With the set of positive real numbers On the product space, and taking values in The set of all continuous functions, here Belonging to this space implies interspecies interaction functions Regarding time and population size Continuous; m is the total number of species in the food chain, i.e., the dimension of the iterative system. (C2) The exponent in the inequality Derived from the cone parameters during the proof process The repeated application of this is used to guarantee the existence condition of the correct solution.
[0035] Secondly, define the function space and the cone.
[0036] set up It is a Banach space. yes A cone in the middle. Assume... and yes An open subset of , and satisfying and Further assume a completely continuous operator. One of the following conditions must be met: (i) When hour, ;when hour, .
[0037] (ii) when hour, ;when hour, .
[0038] Then operator exist There exists a fixed point in it.
[0039] The Guo-Krasnosel'skii fixed-point theorem allows for a global analysis of nonlinear problems on cones, rather than being limited to the properties of local solutions. Using this theorem, this invention can determine the existence and multiplicity of fixed points for nonlinear operators on cones, and can systematically examine the global distribution characteristics and parameter dependencies of solutions under different norm conditions.
[0040] structure Assign a norm to the Banach space: (5) In the formula, For Banach space The norm in is defined as , representing the maximum value of the population size function; Let be a population size function, representing the species' population size over time. Population size.
[0041] consider cone in Defined as: (6) In the formula, The cone parameter, consistent with the time interval partitioning parameter, is used to ensure that the function within the cone remains within the interval. The minimum value on is not lower than the norm. times, that is .
[0042] Define operator as follows: (7) Solve for the Green's function G(t,s), then the operator for: (8) Therefore, the solution to the above iterative system can be expressed as: (9) For ease of calculation, two positive parameters are defined. and Used to determine the range of the growth coefficient: (10) (11) Assuming conditions (C1) and (C2) hold, then for each satisfy (12) In the formula, This serves as the lower bound threshold for the species growth coefficient, ensuring that each species in the food chain has sufficient intrinsic growth momentum to avoid population collapse due to insufficient growth. This is the upper limit threshold for the species growth coefficient, preventing excessive growth of a species from causing food chain imbalances (such as outbreaks of invasive species). As a coordinating factor, falling within this range indicates that the food chain system is in a state of dynamic equilibrium, and the numbers of each species can change in a coordinated manner. The range of values ensures the model's ecological rationality, avoiding unrealistic scenarios such as negative population size or unbounded growth. By adjusting... It can simulate the impact of external pressures such as climate change and habitat destruction on the food chain.
[0043] When all parameters satisfy And boundary values When the value is sufficiently small, it can be proved by applying the Guo-Krasnosel'skii fixed-point theorem that the operator T has at least one fixed point on the cone P, that is, the iterative system has at least one fixed point in equations (1)-(3). Yuan Jie , where m represents the total number of species in the food chain, i.e., the number of trophic levels from producers to apex predators. This theorem is derived by constructing two spheres with different radii. and ,verify The distinct norm compression and expansion conditions are satisfied at the boundary of the cone, thus ensuring that in the toroidal region... The memory is at a fixed point.
[0044] Then, let's set Given by equation (12). Now select the parameters. Make: (13) and, (14) Formula 13 above ensures the recovery potential of endangered species under extreme conditions. If a species' growth coefficient... Below this threshold, the model predicts that the species cannot survive in the food chain (e.g., its recovery rate cannot keep up with its mortality rate due to suppression by predators), ultimately leading to the absence of a system solution (i.e., population collapse). Equation 14 ensures that top predators do not disrupt the balance indefinitely. By limiting... The upper limit of the model ensures that when the population of a species becomes too large, its growth rate will be suppressed by food shortage (limited by its own density) or feedback from natural enemies, thus avoiding the unrealistic situation of "explosive growth of a single species leading to the collapse of the entire food chain".
[0045] Formulas (13)-(14) determine the "life interval" and find the boundary of ecosystem stability. The lower bound of formula (13) prevents system collapse. It specifies... It cannot be too small. If the growth potential of a key species (such as a producer) is too low, it will not be able to recover in time after being preyed upon, and the entire food chain will break down from the bottom up. The upper limit of formula (14): to prevent the system from spiraling out of control. It stipulates... It can't be too big. If a species (such as an invasive species) has too high a growth potential, it will endlessly plunder resources, causing its prey to become extinct, and eventually it will collapse itself.
[0046] We are now seeking fully continuous operators. The fixed point. (By) The definition states that there exists a constant. This makes it possible for each ,have: (15) Pick: (16) set up and ,but Assuming From the properties of the Green's function, we get: (17) In the formula, H 1. To construct the first sphere The radius, i.e. ; In order to make the first Population size function of a species The point in time when the maximum value is reached, i.e. ; H (1 ,sm ) is the Green's function exist The value at that location, i.e. For arbitrarily small positive disturbance parameters ( ); For species The limiting growth rate when the population size approaches zero, i.e. ; The population size function for the first species The norm in the Banach space, i.e.: .
[0047] Similarly, assuming ,but (18) In the formula, In order to make the first - Population size function of a species The point in time when the maximum value is reached, i.e. ; Species The limiting growth rate when the population size approaches zero, i.e. .
[0048] Following this logic, we can conclude that: (19) In the formula, Let t be the value of the population size of the first species at time t.
[0049] definition: (20) In the formula, For Banach space The opening ball is a type of ball played in China, with the origin at its center. , radius is : ; For the Banach space, specifically defined as That is, all defined on closed intervals The set of continuous real-valued functions on the .
[0050] but, (twenty one) Next, by The definition exists This makes it possible for each ,have (twenty two) Using the above formula Scaling the upper and lower limit properties, and letting: (twenty three) Pick and Then there is (twenty four) From the properties of the Green's function, we can obtain... ,have to (25) Similarly, for ,have (26) Similarly, we can obtain (27) definition: (28) but, (29) Applying the Guo-Krasnosel'skii fixed-point theorem to equations (21) and (29), we can see that... exist There exists a fixed point in it. , Indicates from Remove from closure The remaining annular region after the interior of the ring satisfies... All points (if) If it's the kickoff, then the boundary... Included; if If it's a closed ball, attention needs to be paid to the boundary treatment; usually, an open ball is used to avoid overlap. Thus, the positive solutions of formulas (1)-(3) for higher-order iterative systems are obtained. .
[0051] Formulas (17)-(29) construct compression and expansion, thus proving that "coexistence is possible." The compression conditions (Formulas 17, 19) are explained. When the populations of all species remain at a low level At times (such as immediately after a disaster or in the initial stages of protection), the model predicts that their future numbers will not grow indefinitely or fluctuate violently. This indicates that the system is stable and controllable at low densities. Expansion conditions (Equations 25, 27): When the number of species reaches a high level When conditions improve (e.g., recovery is good or an overpopulation trend emerges), the model predicts that its growth will be suppressed (e.g., due to food shortages or increased predators), ensuring that its population does not exceed the carrying capacity of the environment. This demonstrates that the system has a self-regulating ability under high density.
[0052] This embodiment proves the existence of a positive solution to the system using the Guo-Krasnosel'skii fixed-point theorem, which means that under given ecological conditions, all species in the food chain can coexist and maintain a non-negative population, providing computational assurance for the practical application of the model.
[0053] Step S4: Analyze the population changes of each species in the food chain based on the forward solution.
[0054] This embodiment will provide the correct solution. This is considered the ideal baseline state (or "expected state") of the food chain under a given environment. By comparing field-monitored population data with the forward curve, it is possible to diagnose whether the ecosystem has deviated from a healthy trajectory. If the actual population is significantly lower than the forward curve: it indicates that the species may face additional pressures such as overhunting, disease, or habitat degradation, requiring the implementation of conservation measures. If the actual population is significantly higher than the forward curve: it indicates that the species may be experiencing explosive growth (such as an invasive species), disrupting the system balance, requiring the implementation of population control plans.
[0055] Furthermore, the existence of the correct solution depends on a series of parameter conditions, including the growth coefficients of formulas (13) and (14). Limitations. Sensitivity analysis can identify the species or parameters most critical to system stability. Parameters in the model can be artificially adjusted (e.g., ...). , , ), observe when the correct solution disappears or becomes unstable. If If the value is reduced slightly, the correct solution no longer exists, indicating that the growth potential of producers (species 1) is a "weak link" in the system, and their habitat should be prioritized in conservation efforts. If The fact that the value is too high causes the positive solution to disappear, indicating that the potential for the outbreak of the top predator (species 3) is a "risk source" of the system and its population cap needs to be monitored.
[0056] Finally, the forward solution itself provides a stable value for the population size. Combining this with the parameter range, an ecological safety threshold can be derived. Minimum viable population: Under the boundary conditions where the forward solution exists, the corresponding... (Low-level threshold) can be understood as the minimum population size required to maintain system stability. Below this value, the system may not be able to return to equilibrium. Environmental carrying capacity upper limit: the maximum value of the positive solution or... (High-level threshold) can be understood as the maximum population size that the system can withstand without crashing.
[0057] Example 2 In one or more embodiments, a system for analyzing population size in a food chain based on an iterative system is disclosed, specifically including: The data acquisition module is configured to acquire population information in the target food chain. The model building module is configured to: construct a food chain population iterative system for the food chain based on a second-order iterative system with non-homogeneous boundary conditions, wherein the iterative system incorporates predation and predation relationships between species into the equation, and the iterative system has non-homogeneous boundary conditions. The positive solution proof module is configured to: use the Guo-Krasnosel' fixed-point theorem in combination with the eigenvalue range on the Banach space to prove that the iterative system has at least one positive solution; The change analysis module is configured to analyze the population changes of each species in the food chain based on the positive solution.
[0058] Example 3 This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the computer instructions are executed by the processor, they complete the steps of the above-described method for analyzing population size in a food chain based on an iterative system.
[0059] Example 4 This embodiment provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the above-described method for analyzing population size in a food chain based on an iterative system.
[0060] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0061] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0062] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment, whereby a series of operational steps are performed to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0063] The descriptions of each embodiment in the above embodiments have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.
[0064] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for analyzing population size in a food chain based on an iterative system, characterized in that, include: Obtain population information within the target food chain; A food chain population iterative system is constructed based on a second-order iterative system with non-homogeneous boundary conditions. The iterative system incorporates the predator-prey relationship between species into the equation and has non-homogeneous boundary conditions. By using the Guo-Krasnosel' fixed-point theorem and the range of eigenvalues in the Banach space, it is proved that the iterative system has at least one positive solution; Based on the positive solution, the population changes of each species in the food chain are analyzed.
2. The method for analyzing population size in a food chain based on an iterative system as described in claim 1, characterized in that, The second-order iterative system with non-homogeneous boundary conditions is expressed as: In the formula, Let be the population size of the i-th species in the food chain at time t; The second-order rate of change of population size; This refers to environmental factors or the intrinsic growth coefficient of a population. Let be the interspecies interaction function, representing the influence of the growth rate of the i-th species on the growth rate of its next trophic level. The impact; To indicate that the food chain is a closed loop; m is the total number of species in the food chain.
3. The method for analyzing population size in a food chain based on an iterative system as described in claim 2, characterized in that, The non-homogeneous boundary conditions are as follows: In the formula, Let i be the initial population size of species i at the start of time. Let i be the target population size of species i at the end of time.
4. The method for analyzing population size in a food chain based on an iterative system as described in claim 1, characterized in that, The proof process includes constructing... Assign a norm to the Banach space: In the formula, For Banach space Norms in; The population size function; structure cone in for: In the formula, For cone parameters; The operator is constructed as follows: In the formula, Let i be an operator, i = 1, ..., m; For producers; By applying the Guo-Krasnosel'skii fixed-point theorem, we prove that the operator T has at least one fixed point on the cone P.
5. The method for analyzing population size in a food chain based on an iterative system as described in claim 4, characterized in that, The limiting growth rate parameter is constructed as follows: In the formula, When the population size is close to zero, it indicates that the species is endangered. When the population size approaches infinity, it represents a state of species explosion. This is a time sub-interval.
6. The method for analyzing population size in a food chain based on an iterative system as described in claim 5, characterized in that, The range of the growth coefficient is determined using positive parameters: Assuming the following conditions hold, then for each satisfy In the formula, is the lower bound threshold for the species growth coefficient; m is the total number of species in the food chain; This is the upper bound threshold for the species growth coefficient; It is the set of positive real numbers; It is a continuous function space.
7. The method for analyzing population size in a food chain based on an iterative system as described in claim 4, characterized in that, The Guo-Krasnosel'skii fixed-point theorem is applied to prove that the operator T has at least one fixed point on the cone P. Specifically, this is done by constructing two spheres with different radii. and This verifies that operator T satisfies different conditions for norm compression and expansion on the boundary of the cone, thus enabling it to function in the annular region. The memory is at a fixed point.
8. A system for analyzing population size in a food chain based on an iterative system, characterized in that, include: The data acquisition module is configured to acquire population information in the target food chain. The model building module is configured to: construct a food chain population iterative system for the food chain based on a second-order iterative system with non-homogeneous boundary conditions, wherein the iterative system incorporates predation and predation relationships between species into the equation, and the iterative system has non-homogeneous boundary conditions. The positive solution proof module is configured to: use the Guo-Krasnosel' fixed-point theorem in combination with the eigenvalue range on the Banach space to prove that the iterative system has at least one positive solution; The change analysis module is configured to analyze the population changes of each species in the food chain based on the positive solution.
9. An electronic device, characterized in that, The method includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, perform the method for analyzing population size in a food chain based on an iterative system as described in any one of claims 1-7.
10. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, perform the method for analyzing population size in a food chain based on an iterative system as described in any one of claims 1-7.