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Lotka-Volterra Equation

作者: AnonTokyo

简介: Differential equation system describing prey and predator population change through time.

最后修改: 2025-04-08 09:09:09.956494

文章状态: 已发布

标签:

生物 数学 编程
In tro duction Understanding the dynamics b et w een predator and prey p opulations is a fundamen tal asp ec t of eco- logical researc h. In this study , w e in v e stigate ho w c hanges in prey p opulation size inuence the dynamics of predator p opulations within an insulated en vironmen t with limited resources. By sim ulating this scenario using Python, w e aim to exp lore the in tricate relationship b et w een predator and prey p opulations in a con- trolled en vironmen t. This r e searc h question i s crucial for gain ing insigh ts in to the mec han ism s that go v ern predator-prey in teractions and can ha v e signican t implications for ecosystem managemen t and conserv a- tion eorts. Through our exp erimen tal sim ulation, w e seek to shed ligh t on the impact of prey p opulation uctuations on predator p opulations and con tribute to the broader understanding of ecological dynamics. Researc h Question Ho w do p re y p opulation aect predator p opulation under a insulated en vironmen t with limited re- sources? Bac kground Researc h The Lotk a-V olterra equation, al s o kno wn as the predator-prey mo del, is a foundational concept in eco- logical and mathematical biology that describ es the dynamics b et w een t w o sp ecies: one as a predator and the ot her as its prey (Stern b e r g, 2009). [ 4 ] The mo del, w h ic h w as separately dev elop ed b y Vito V olterra in 1926 and Alfr e d J. Lotk a in 1925, sho ws ho w the p opulations of the predator and prey aect one another o v er time through a system of rst-order non-linear dieren tial equations. American bioph ysicist Lotk a rst utilized these equati ons to describ e c hemical reactions, whereas Italian mathematic i an V olterra conc en- trated on ecological in teractions, esp ecially to explain the oscillatory p opulation shifts seen in shing data. Since th e n , the equations ha v e b een us ed extensiv ely in the study of ecological systems to illustrate ideas lik e p opulation cycles, stabili t y , and ho w c hanges in the en vironmen t aec t in teractions b et w een dieren t sp ecies. The Lotk a-V ol te r ra mo del oers a vital theoretical framew ork for comprehending in tricate biological in teractions despite its idealized assumptions and simplicit y , and it has spark ed a great deal of dev elopmen ts and applications in a wide range of scien tic domain s . Lotk a in v estigates the e x is tence of rh ythmic eects in c hemical reactions, p oin ting out that p revious researc h frequen tly disco v ered that these oscillations w ere mild and eeting. Con trary to earlier theories, Lotk a disco v ers via his o wn researc h the circumstance s in whic h un damp ed, p ermanen t oscillations can o ccur. He mo dels the ev oluti on of systems made up of dieren t ki nds of matter using dieren tial equations, taking in to accoun t b oth biological en tities and elemen ts of the inorgani c en vironmen t. Prediction The p opulation of prey and predator will b e in a cyclical mismatc h relationship. Th e n um b er of prey increases and decreases in a p erio d, and the n um b er of predators follo ws th e same p erio d b ut with a dieren t phase. This is b ecause p erio dic phenomena pla y an imp ortan t role i n natur e , where p ermanen t osc ill ations often o ccur in natural systems(Chasno v, 2022) . [ 1 ] 1
Mo del Establishmen t Assumption F ew assumptions will ha v e to b e made to ensure accuracy of the mo del: • The natural system consists of only t w o sp ecies: Prey and Predator • The natural system is close d • Predator’s only f o o d source is prey • The natural system ha v e sucien t resource to supp ort Prey and Predator • Mortalit y rate and p opulation grw oth rate are constan t • Probabilit y of a success P redation is constan t • Prey’s abilit y to feed predator is constan t V ariable Description V ariable Unit Descritpion t mon th Time x N/A P opu lation of Prey y N/A P opu lation of Predator r N/A P opulation gro wth rate of Prey a N/A Probabilit y of a success Predation d N/A Mortali t y rate of Prey b N/A Prey abilit y to feed predator T able 1: V ariable T able Lotk a-V olterra Equation The indep enden t p opulation gro wth for prey can b e denoted b y dieren tial equation ˙ x = r x After in tro du c i ng Predator to n atural system, Predator allo ws the prey gro wth rate to decrease, where th e decrease is prop ortional to Predator p opulation (y) r → r − ay The indep enden t p opulation gro w for predator can b e denoted b y dieren tial equation ˙ y = − dy 2
After in tro du c i ng Prey to natur al system, Prey allo ws the mortalit y rate of Predator to decrease, where th e decrease is prop ortional to Prey p opulation (x) d → d − bx Therefore, w e could deriv e the Lotk a-V olterra Equation (Lotk a, 1920) [ 2 ]: ˙ x = ( r − ay ) x (1) ˙ y = − ( d − bx ) y (2) Mo del Solving This set of dieren tial equation ha v e no analytic solution , th us w e use p ython to sim ulate a n umerical solution. The sim ulation runs under the follo wing conditions: r = 1 a = 0.1 d = 0.5 b = 0.02 The initial cond ition is giv en as: x(t = 0) = 25 y(t = 0) = 2 Metho d applied to sim ulate n umerical solution is rst-order diere n tial appr o ximation with a ste p length of 0.001 f ( x + dx ) ≈ f ( x ) + f ′ ( x ) dx Result Analysis The sim ulation results are sho wn b elo w: Figure 1: P opulation of Prey and Predator plotted with resp ect to time 3
Figure 2: P opulation of Prey plotted with resp ect to Predator As can b e obtained from Figure 1, the p eri o d of the t w o functions is ar ound 10.8. Using the n u m erical in tegration metho d, it can b e obtained that the a v erage v alues in a p erio d for Prey is 25 and for Predator is 10, resp ectiv ely . As can b e seen from Figure 2, the relationship b et w ee n x and y is in the f orm of a closed function whic h can b e solv ed mathem ati c all y as an implicit function b y eliminating t. [ 3 ] Divide the t w o dieren tial equation to get: dx dy = x ( r − ay ) y ( − d + bx ) Separating the v ariables and in tegrating b oth sides giv e s: − dl n ( x ) + bx = r l n ( y ) − ay + c 1 Simplify , and w e get the n u me ri c al relationship b et w een p opulation of Predator and Prey . This is dened as the phase tra jec t ory of the system of dieren tial equations. ( x d e − bx )( y r e − ay ) = c (3) Where constan t ”c” is determined b y initial condi tion. In our condition, c ≈ 4.97 Observing b oth t w o gure w e nd that the Predator and Prey p op ulations do es not reac h an equilib- rium p oin t. But from the dieren tial equation itself, w e can deriv e an equilibrium p oin t with zero rate of c hange. Ob viously , when x = d b and y = r a , the system reac hes equilibrium. 4
Figure 3: Prey-Predator graph sim ulated with dieren t initial condition If w e plot this ordered-pair in to graph and sim ulate a few more grap hs with die r e n t initial condition as sho wn in Figure 3, w e nd that p oin t p is the cen ter of this family of phase tra jectories. Th us, in an y initial v alue that do es not satisfy th e unique equilibrium cond ition, the phase tra j e ctory is a closed path motion around p oin t p. Mo del Ev aluation The mo del has sev eral limitations: • The phase tra jectory is a closed curv e and the structure lac ks stabilit y . After c hanging the initial v alue it en ters another curv e from whic h there is no reco v ery . In nature, ecosystems with p erio dic equilibrium are structurally stable and ha v e in ternal c on s tr ain ts to restore the system to it s original state after deviating from its tra jectory . • Mo deling assumptions are to o limited to accommo date more complex fo o d w ebs and ecological rela- tionships in nat ure. • Mo del do es not consider comp etition within sp ecies p opulations. • Mo del do es not consider en vironmen tal carry ing capacit y . • Mo del do es not consider the predator’s p redation preference to the exten t that it prev en t s the pr e y p opulation from repro ducing ecien tly when it consumes to o m uc h of a single-sex prey . • Mo dels do es not consider c hanges in ge n e ti c frequency caused b y selectiv e pressure whic h migh t eec t p opulation surviv al or predatory capacit y . Conclusion By sim ulating the n umerical s ol ution of Lotk a-V olterra Equation, w e conclude that the prey and preda- tor p opulati on size s sho w p eri o dic uctuations o v e r time . Both uctuate in similar cycles, but with some 5
phase dierences. Therefore, the p erio dic staggere d uctuation relationship predicted at the b eginning of the pap er is v eri ed b y the mo del. If the time v ariable t is eliminated b y dividing the t w o equations, the prey and predator p opulations are m utually explicit, resulting in a closed cur v e cen tered on the equili brium p oin t. The curv e carries a p ending parameter determined b y the initial conditions. Dieren t initial conditions can determine diere n t closed curv es, but since the e q uilibrium p oin t remains the same, the cen ter remains the same. AI Usage Clarication Clarication: In preparation of this w ork, the author used Chat-GPT 3.5 to create the text con ten t. After using th is to ol, the author review ed and edited the con ten t as needed. Input Prompt 1: write me a paragraph ab out bac kground rese arc h on The Lotk a - V olte r ra equation Output 1: The Lotk a-V olterra equati on, also kno wn as the predator-prey mo del, is a foundational concept in ecolog- ical and mathematical biology that describ es the dynamics b et w een t w o sp ecies: one as a predator and the other as its prey . Dev elop ed indep enden tly b y Alfr e d J. Lotk a in 1925 and Vito V olterra in 1926, th e mo del uses a set of rst-order, non-linear dieren tial equations to illustrate ho w the p opulation sizes of the predator and prey inuence eac h other o v er time . Lotk a, an American bioph ysicist, applied these equations to c hemical reactions initially , whil e V olterra, an Italian mathematician, fo cused on ecological in teractions, particularly to explain oscillatory p opulation c hanges observ ed in shery data. The equations ha v e s in c e b ecome instrumen tal in studying e cological systems, illustrating concepts suc h as p opulation cycles, stabil- it y , and the impact of en vironmen tal c hanges on sp ecies in teractions. Despite its s imp licit y and idealized assumptions, the Lotk a-V olterra mo del pro vides a crucial theoretical framew ork for understand ing complex biological in teractions and has inspired n umerous extensions and applications across v ari ous scien tic elds. Input Prompt 2: write me a literature review u p on th is eassa y , mak e it short Output 2: Alfred J. Lotk a’s 1920 pap er, ”Analytical Note on Certain Rh yt hmic Relations in O rganic Systems,” ex- plores p erio dic phenomena in organic and inorganic systems through the lens of mathematical mo deling. This literature review summarizes Lotk a’s approac h, nd ings, and con tributions to the eld. Bac kground and Con text Lotk a builds on pr e v ious w ork b y researc hers suc h as M c E w en and Mic hael (1919), Reed and Holland (1919), and Rob e r ts on (1913), who explored v arious biological and c hemical dynamics using statistical and analytical metho ds. Lotk a ac kno wledges the supp ort of Dr. G.F. McEw en, highligh ting the collab orativ e nature of his researc h. Key Concepts and Metho d ology Lotk a examines the o cc u rrence of rh ythmic ee cts in c hemical reactions, noting that earlier studies often found these oscillations to b e damp ed and transien t. Ho w ev er, through his o wn in v estigations, Lotk a iden ties conditions under whic h un damp ed, p ermanen t oscillations can arise, con trary to previous assumptions. He uses dieren tial equations to mo del the ev olution of systems comprising v arious sp ecies of matter, considering b oth biological or ganisms and comp onen ts of the inorganic en vi ronmen t. 6
Reference [1] Jerey R. Chasno v . 1.4: The lotk a-v olterra predator-prey mo del, 01 2022. [2] Alfred J. Lotk a. Analytical note on certain rh ythmic relations in organic systems. Pr o c e e dings of the National A c ademy of Scienc es , 6:410–415, 06 1920. [3] Joseph M. Mahay and San Diego State Univ ersit y . Lotk a-v olterra mo dels, 2010. [4] Shlomo Stern b erg. Lecture 15 lotk a- v olterra, 04 2009. 7
App endix i m p o r t m a t p l o t l i b . p y p l o t a s p l t i m p o r t n u m p y a s n p i m p o r t m a t h # d x / d t = ( r − a y ) x p r e y , C h a n g e s i n t h e s i z e o f p r e y p o p u l a t i o n s r e s u l t f r o m g r o w t h a t t e n u a t i o n d u e t o p r e d a t i o n o n t h e m s e l v e s # d y / d t = − ( d − b x ) y p r e d a t o r , C h a n g e s i n t h e s i z e o f t h e p r e d a t o r c o m m u n i t y i s t h e g r o w t h o f t h e p r e d a t o r c o m m u n i t y , m i n u s i t s n a t u r a l d e a t h s x 0 = 2 5 y 0 = 2 t s t e p = 0 . 0 0 1 t 0 = 0 r = 1 # p r e y p o p u l a t i o n g r o w t h r a t e a = 0 . 1 # m e a s u r e p r e d a t o r a b i l i t y t o p r e d a t e o n p r e y d = 0 . 5 # p r e d a t o r m o r t a l i t y r a t e b = 0 . 0 2 # m e a s u r e p r e y a b i l i t y t o f e e d p r e d a t o r c o n s = m a t h . p o w ( x 0 , d ) ∗ m a t h . e x p ( − b ∗ x 0 ) ∗ m a t h . p o w ( y 0 , r ) ∗ m a t h . e x p ( − a ∗ y 0 ) p r i n t ( c o n s ) x = [ ] y = [ ] t = [ ] # S i m u l a t i o n b y F i r s t − o r d e r d i f f e r e n t i a l a p p r o x i m a t i o n d e f s i m u l a t i o n X ( x , y , r , a , t s t e p ) : r e t u r n x + ( r ∗ x − a ∗ x ∗ y ) ∗ t s t e p d e f s i m u l a t i o n Y ( x , y , b , d , t s t e p ) : r e t u r n y + ( − d ∗ y + b ∗ x ∗ y ) ∗ t s t e p d e f m a i n ( x 0 , y 0 , t s t e p , t 0 , r , a , b , d , t a r g e t ) : t = t 0 x = x 0 y = y 0 x l i s t = [ x 0 ] y l i s t = [ y 0 ] t l i s t = [ t 0 ] w h i l e t < t a r g e t : x , y = s i m u l a t i o n X ( x , y , r , a , t s t e p ) , s i m u l a t i o n Y ( x , y , b , d , t s t e p ) x l i s t . a p p e n d ( x ) y l i s t . a p p e n d ( y ) t l i s t . a p p e n d ( t ) 8
t + = t s t e p r e t u r n x l i s t , y l i s t , t l i s t # C a l c u l a t e p e r i o d d e f f i n d P e r i o d ( v l i s t , t l i s t ) : c o u n t = 0 m l i s t = [ ] f o r j i n t l i s t : i f c o u n t = = 0 : c o u n t + = 1 e l i f v l i s t [ c o u n t − 1 ] < v l i s t [ c o u n t ] a n d v l i s t [ c o u n t + 1 ] < v l i s t [ c o u n t ] : m l i s t . a p p e n d ( j ) c o u n t + = 1 e l i f c o u n t + 1 = = l e n ( v l i s t ) : d i f f = [ m l i s t [ i + 1 ] − m l i s t [ i ] f o r i i n r a n g e ( l e n ( m l i s t ) − 1 ) ] p r i n t ( d i f f ) r e t u r n d i f f [ 0 ] c o u n t + = 1 # C a l c u l a t e A v e r a g e d e f f i n d A v e r a g e ( p e r i o d , v l i s t ) : p e r i o d / = 0 . 0 0 1 p e r i o d = r o u n d ( p e r i o d ) s u m = 0 c o u n t = 0 f o r i i n v l i s t : i f c o u n t = = p e r i o d : r e t u r n s u m / p e r i o d e l s e : s u m + = i c o u n t + = 1 # p l o t f i g 2 x , y , t = m a i n ( x 0 , y 0 , t s t e p , t 0 , r , a , b , d , t a r g e t = 1 5 ) p l t . p l o t ( x , y ) p l t . x l a b e l ( ” P r e y   P o p u l a t i o n ” ) p l t . y l a b e l ( ” P r e d a t o r   P o p u l a t i o n ” ) p l t . t i t l e ( ” R e l a t i o n s h i p   B e t w e e n   P r e y   a n d   P r e d a t o r   P o p u l a t i o n ” ) p l t . g r i d ( ) p l t . s h o w ( ) 9
# p l o t f i g 1 p l t . p l o t ( t , x , l a b e l = ” P r e y   P o p u l a t i o n ” ) p l t . p l o t ( t , y , l a b e l = ” P r e d a t o r   P o p u l a t i o n ” ) p l t . l e g e n d ( ) p l t . x l a b e l ( ” T i m e ” ) p l t . y l a b e l ( ” P o p u l a t i o n ” ) p l t . t i t l e ( ” R e l a t i o n s h i p   B e t w e e n   P r e y   a n d   P r e d a t o r   P o p u l a t i o n   T h r o u g h   T i m e ” ) p l t . g r i d ( ) p l t . s h o w ( ) # f i n d p e r i o d x p e r i o d = f i n d P e r i o d ( x , t ) y p e r i o d = f i n d P e r i o d ( y , t ) # f i n d a v e r a g e p r i n t ( f i n d A v e r a g e ( x p e r i o d , x ) ) p r i n t ( f i n d A v e r a g e ( y p e r i o d , y ) ) # p l o t m u l t i p l e p h a s e t r a j e c t o r y x , y , t = m a i n ( x 0 , y 0 , t s t e p , t 0 , r , a , b , d , t a r g e t = 1 5 ) p l t . p l o t ( x , y ) f o r i i n r a n g e ( − 1 , 7 ) : x , y , t = m a i n ( x 0 + i , y 0 + i , t s t e p , t 0 , r , a , b , d , t a r g e t = 1 5 ) p l t . p l o t ( x , y ) p l t . s c a t t e r ( d / b , r / a ) p l t . a n n o t a t e ( ” P ” , x y = ( d / b , r / a ) , x y t e x t = ( d / b + 0 . 5 , r / a + 0 . 5 ) ) p l t . x l a b e l ( ” P r e y   P o p u l a t i o n ” ) p l t . y l a b e l ( ” P r e d a t o r   P o p u l a t i o n ” ) p l t . t i t l e ( ” R e l a t i o n s h i p   B e t w e e n   P r e y   a n d   P r e d a t o r   P o p u l a t i o n ” ) p l t . g r i d ( ) p l t . s h o w ( ) 10
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